New Mexico Statutes 38-1-3 provides:
In all the courts in this state the common law as recognized in the United States of America, shall be the rule of practice and decision.

Petition of Variable for Change of Name v. Nash, 190 P. 3d 354 - NM: Court of Appeals 2008 provides:
Petitioner has a right under the common law to assume any name that he wants so long as no fraud or misrepresentation is involved. In re Ferner, 295 N.J.Super. 409, 685 A.2d 78, 80 (Ct. Law Div.1996); In re Rivera, 165 Misc.2d 307, 627 N.Y.S.2d 241, 244 (Civ.Ct.1995). He may do so without making any application to the state. Thus, under the common law, Petitioner may exercise his right to free speech and use any name at all.

IN THE MATTER OF PETITION OF MOKILIGON, 2005 NMCA 21 - NM: Court of Appeals 2005 states:
In that case, in which the court reversed the district court's denial of the petitioner's request to change his name to Santa Claus, the court noted that name change statutes "merely provide a codified process to aid an individual's common law right to adopt another name at will." Id. And the court emphasized that, "[c]onsequently, applications under the statute should be encouraged and should generally be granted unless sought for a wrongful or fraudulent purpose." Id. (internal quotation marks and quoted authority omitted).