It might be tempting to think of compression, especially in an east-west direction. However at the south end, New Mexico is experiencing east-west extension (grabens including Rio Grande Rift Valley, plus associated volcanism).
Hot spot and/or mantle plume uplift doesn't appear to explain them either. Yes there's the Yellowstone Hotspot under NW Wyoming. Regardless of whether this is really caused by a mantle plume or not, the related volcanism does not correlate with the mountains.
Just one of many pictures I found: