Your button's emblem is the distinctive crown of the Prussian (German) Empire. Your button dates from the late-1800s through 1918, when the Prussian Empire ended after its defeat in World War One. Your button was worn on Prussian Empire army enlisted-men's uniforms. (Officers' buttons had a different emblem.)
Although it seems surprising to find one in New Mexico, others have come from Southwestern US states. Perhaps WW1 German prisoners were held there. Or, yours may have been a souvenir taken home by a WW1 US soldier.