Even in deserts, you rarely need a compass, because you are surrounded by obvious landmarks, at least in the continental US. You will need a compass for navigation when your visibility is restricted (fog, typically) and if you are at sea. I can count on one hand the times I have seriously used a compass in the mountains, and both times we were in the clouds.

But if you are going to carry a compass, carry one that is marked in at least two degree increments and that has a sighting (signal) mirror. Plenty are around that weigh maybe an ounce. Very easy to carry. If you use it much, you will want it on a neck lanyard., and not in your pockets.