In Arizona or New Mexico, what is the altitude below which you would expect most or all winters be frost-free?

Answer 1

Below 2000 feet, frosts are uncommon. Here's some more information about the definition of the low desert zone:

LOW ALTITUDE DESERT (Elevation 1000-2,000 feet) Zone 5 growing seasons are long and the average minimum winter temperature recorded for this zone is around 36-37 degrees. However, temperatures occasionally dip below 20 degrees. In summer, average maximum temperatures are near 102 degrees. Annual rainfall is ten inches or less throughout Zone 5.