Exhibit 99.4

October 7, 2016

The Board of Directors

AmSurg Corp.

1A Burton Hills Boulevard

Nashville, TN 37215

Re: Amendment No. 2 to Registration Statement on Form S-4 of New Amethyst Corp. (“New Amethyst”) filed on October 7, 2016 (the “Registration Statement”)

Members of the Board:

Reference is made to our opinion letter, dated June 15, 2016 (the “Opinion Letter”) relating to the Mergers (as defined in the Registration Statement).

We hereby consent to the inclusion of the Opinion Letter as Annex C to the Joint Proxy Statement/ Prospectus of New Amethyst, which is a part of the Registration Statement, and to the description of the Opinion Letter and the references to our name contained therein under the headings “SUMMARY—Opinions of AmSurg’s Financial Advisors—Opinion of Guggenheim Securities, LLC,” “THE MERGERS—Background of the Mergers,” “THE MERGERS—AmSurg’s Reason for the Mergers; Recommendation of the AmSurg Board” and “THE MERGERS—Opinion of AmSurg’s Financial Advisor—Opinion of Guggenheim Securities.”

In giving the foregoing consent, we do not admit that we come within the category of persons whose consent is required under Section 7 of the Securities Act of 1933, as amended (the “Securities Act”), or the rules and regulations promulgated thereunder, nor do we admit that we are experts with respect to any part of such Registration Statement within the meaning of the term “expert” as used in the Securities Act or the rules and regulations promulgated thereunder.

Very truly yours,

/s/ Guggenheim Securities, LLC

GUGGENHEIM SECURITIES, LLC