INVESTOR RIGHTS AGREEMENT

Exhibit 4.1

Investor Rights Agreement

Dated as of October 13, 2005

among

Energy XXI Acquisition Corporation (Bermuda) Limited

Sunrise Securities Corp.

and

Collins Stewart Limited

INVESTOR RIGHTS AGREEMENT

This Investor Rights Agreement (this “Agreement”) is made and entered into as of 13 October 2005 by and among Energy XXI Acquisition Corporation (Bermuda) Limited, a Bermudan company (the “Company”), and Sunrise Securities Corp. and Collins Stewart Limited (collectively, the “Placement Agents”), for the benefit of the holders (the “Holders”) from time to time of the Company’s common shares, par value $.001 per share (the “Common Shares”) and warrants (the “Warrants”), each Warrant for the purchase of one (1) Common Share.

Reference is made to the Company’s Offering Circular, dated 13 October 2005 (the “Offering Circular”), relating to the offer and sale of fifty million (50,000,000) units (the “Units”), each Unit consisting of one (1) Common Share and two (2) Warrants. For the benefit of the Holders and in consideration of the Placement Agents entering into a placing agreement with the Company for the placing of the Units, the Company has agreed to provide the investor rights set forth in this Agreement.

The parties hereby agree as follows:

SECTION 1. DEFINITIONS

As used in this Agreement, the following capitalized terms shall have the following meanings:

ADR: As defined in Section 2(b) hereof.

Commission: The U.S. Securities and Exchange Commission.

Exchange Act: The U.S. Securities Exchange Act of 1934, as amended.

NASD: National Association of Securities Dealers, Inc.

Offering: The offering of the Units pursuant to the Offering Circular.

Person: An individual, partnership, corporation, limited liability company, unincorporated organization, association, joint-stock company, trust, joint venture, government or any agency or political subdivision thereof or any other entity.

Placing Agreement: The placing agreement between the Company, Sunrise Securities Corp., Collins Stewart and the Company’s directors dated on or about the date of this Agreement.

Qualified Business Combination: A business combination which, when combined with the Company’s previous business combinations, results in the release of at least fifty percent (50%) of the initial amount of the trust fund (as more fully described in the Offering Circular).

Securities Act: The U.S. Securities Act of 1933, as amended.

‘34 Act Liquidated Damages: As defined in Section 5(a) hereof.

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‘34 Act Registration Deadline: The date of consummation of a Qualified Business Combination or, alternatively, in the case where no Qualified Business Combination shall have occurred but a business combination shall have been completed and the balance of the funds in the trust fund have been distributed to the new shareholders, the date of such distribution.

‘34 Act Registration Default: As defined in Section 4(a) hereof.

‘34 Act Registration Statement: A registration statement of the Company on Form 10 (or such other form which it is appropriate to use to register the Common Shares or ADRs under the Exchange Act), including all amendments and supplements thereto and all exhibits and material incorporated by reference therein.

SECTION 2. EXCHANGE ACT REGISTRATION AND LISTING

(a) The Company shall (i) no later than the date that is one hundred eighty (180) days following the ‘34 Act Registration Deadline cause to be filed with the Commission a ‘34 Act Registration Statement, and (ii) use its commercially reasonable efforts to cause such ‘34 Act Registration Statement to be declared effective on or prior to the date that is two hundred seventy (270) days following the ‘34 Act Registration Deadline.

(b) As promptly as practicable after the ‘34 Act Registration Statement shall have been declared effective, the Company shall use its commercially reasonable efforts to cause the Common Shares or American Depositary Receipts representing Common Shares (“ADRs”) to be authorized to be quoted and/or listed (to the extent applicable) on the American Stock Exchange, the New York Stock Exchange, the National Association of Securities Dealers, Inc. Automated Quotation System or the Nasdaq National Market (or, in each case, a successor thereto) or a similarly recognized national trading platform, if the Common Shares or ADRs so qualify.

SECTION 3. WARRANT SHARE REGISTRATION

(a) The Company shall file with the Commission, no later than the 60th day after the ‘34 Act Registration Statement is declared effective, a shelf registration statement (the “Shelf Registration Statement”) pursuant to Rule 415 under the Securities Act of 1933, as amended (the “Securities Act”) covering the Common Shares as and when issued upon exercise of the Warrants, including the Warrants underlying that certain unit purchase option dated October 13, 2005 (“Purchase Option”) issued by the Company to Sunrise Securities Corp. and/or its designees (the “Warrant Shares”). The Company shall (i) use commercially reasonable efforts to cause the Shelf Registration Statement to be declared effective under the Securities Act no later than 105 days after the ‘34 Act Registration Statement is declared effective unless the Commission reviews such registration statement in which case the 105 day period shall be increased to 120 days, and (ii) use commercially reasonable efforts to keep the Shelf Registration Statement effective until the earlier of (x) the date when the holders of the Warrants and the Warrant Shares are able to sell such shares immediately without restriction pursuant to volume limitation provisions of Rule 144 under the Securities Act; and (y) such time as all of the Warrant Shares have been sold either pursuant to the Shelf Registration Statement or pursuant to Rule 144 under the Securities Act or any similar provision then in force.

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(b) The Company shall use commercially reasonable efforts to cause the prospectus contained within the Shelf Registration Statement to be usable for its intended purpose during the Designated Period (as defined below) including filing any amendments and supplements to the Shelf Registration Statement or the prospectus as may be required. The “Designated Period” shall mean the period beginning two days after the public announcement of quarterly and year-end earnings by the Company and ending 30 days thereafter, provided that the relevant earnings announcement shall in no event be issued publicly after November 29, March 1, May 30 and October 28, in the case of the first quarter, second quarter, third quarter and fiscal year-end, respectively.

(c) The Company may suspend the effectiveness of the Shelf Registration Statement or the use of the prospectus that is part of the Shelf Registration Statement during specified periods during the Designated Period under certain circumstances relating to pending corporate developments, public filings with the Commission and similar events; provided, that due to concerns about possibly disseminating material non-public information, the Company need not specify the nature of the event giving rise to a suspension in any notice to holders of the Warrant Shares of the existence of such a suspension. Any such suspension period shall not exceed an aggregate of 45 days of the Designated Periods in any 360-day period. The number of days during which the Company may have suspended use of the prospectus during the Designated Periods will be added to the end of the Designated Period with respect to the next one or more successive quarterly periods or, when the prospectus has been appropriately amended or supplemented to reflect the event or events giving rise to a suspension, the amended or supplemented prospectus shall be delivered to the selling Holders named in the Shelf Registration Statement and they shall have a period of time equal to the number of days during which selling was suspended to sell Warrant Shares using the amended or supplemented prospectus.

(d) In connection with the preparation of the Shelf Registration Statement, the Company shall mail a questionnaire to all warrantholders and holders of restricted Warrant Shares of record as of the close of business on the day the ‘34 Act Registration Statement becomes effective. In order to be named as a selling shareholder in the prospectus at the time of effectiveness of the Shelf Registration Statement, such Holder must complete and deliver the questionnaire to the Company on or prior to the 15th business day after the questionnaire is mailed. Holders not named as selling shareholders therein will not be able to avail themselves of such Shelf Registration Statement. Accordingly, transferees of warrants or Warrant Shares may not be able to avail themselves of registration; provided, that, every six months after the Shelf Registration Statement becomes effective, the Company shall mail to all warrantholders of record and holders of restricted Warrant Shares of record a new questionnaire and amend or supplement the Shelf Registration Statement to include those that have duly completed and returned a questionnaire to the Company. The Company may, in its sole discretion, permit more frequent supplements to the Shelf Registration Statement to include transferees. Failure to timely provide a duly completed questionnaire will result in the non-responding security holders exclusion from the Shelf Registration Statement or the use of the prospectus that is part of the Shelf Registration Statement. Any Holder who has properly elected to be included in the Shelf Registration Statement will (i) be required to be named as a selling stockholder in the related prospectus, (ii) be required to deliver a prospectus to purchasers, and (iii) be subject to the civil liability provisions under the Securities Act in connection with any sales pursuant to such Shelf Registration Statement.

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SECTION 4. “PIGGYBACK” REGISTRATIONS

(a) If at any time the Company shall propose to register under the Securities Act (other than pursuant to Sections 2 and 3 of this Agreement) any of its Common Shares for the account of its Existing Shareholders (as defined in the Offering Circular), it will promptly give written notice to all Holders of Common Shares that are (i) holders of Common Shares bearing a restrictive legend that are not freely saleable in the United States under Rule 144(k) of the Securities Act, and (ii) that are known by the Company to be affiliates of the Company (other than officers and directors of the Company) (such Common Shares hereinafter referred to as “Registrable Shares”) of its intention so to do. Upon the written request of any such holder, received by the Company within 20 days after the giving of any such notice by the Company, to register any or all of its Registrable Shares, the Company will use its reasonable best efforts to cause the Registrable Shares as to which registration shall have been so requested to be included in the securities to be covered by the registration statement proposed to be filed by the Company, all to the extent requisite to permit the sale or other disposition by the holder (in accordance with its written request) of such Registrable Shares so registered.

(b) If the registration of which the Company gives notice as provided above is for a registered public offering involving an underwriting, the Company shall so advise the Holders of Registrable Shares as a part of the written notice given pursuant to this Section 4. In such event, the right of any Holder of Registrable Shares to registration pursuant to this Section 4 shall be conditioned upon such Holder’s participation in such underwriting to the extent provided herein. All Holders of Registrable Shares proposing to distribute their securities through such underwriting shall (together with the Common Shares to be registered by the Company in such registration (the “Other Shareholders”)) enter into an underwriting agreement in customary form with the underwriter or underwriters selected for underwriting by the Company. If any Holder of Registrable Shares disapproves of the terms of any such underwriting, it may elect to withdraw therefrom by written notice to the Company and the underwriter. Any Registrable Shares or other securities excluded or withdrawn from such underwriting shall be withdrawn from such registration.

(c) Notwithstanding any other provision of this Section 4, if the underwriter determines that marketing factors require a limitation on the number of shares to be underwritten, the underwriter may exclude from such registration and underwriting all or a portion of the Registrable Shares which would otherwise be underwritten pursuant to this Section 4. The Company shall so advise all Holders of securities requesting registration of any limitations on the number of shares to be underwritten and the number of shares of securities that are entitled to be included in the registration; provided, that the number of shares to be underwritten shall be allocated pro rata among all Holders of Registrable Shares and the Existing Shareholders in proportion, as nearly as practicable, to the respective amounts of Registrable Shares owned by them.

SECTION 5. LIQUIDATED DAMAGES

(a) If the ‘34 Act Registration Statement shall not have been declared effective on or prior to the date that is two hundred seventy (270) days following the ‘34 Act Registration Deadline (a “‘34 Act Registration Default”), the Company shall, as promptly as practicable and in no event later than two (2) days following the end of the month in which such ‘34 Act Registration Default occurred, pay to each Holder of Common Shares liquidated damages (“‘34 Act Liquidated Damages”) in the form of Common Shares in an amount equal to 0.5% of the number of such Holder’s Common Shares held on the date of such ‘34 Act Registration Default; provided, that a Holder shall be paid ‘34 Act Liquidated Damages only with respect to (i) any Common Shares that were acquired by such Holder upon the