As many have already stated, a compass on the moon would work rather poorly due to the fact that the moon has a very weak magnetic field. The compass that you have may move slightly or just not at all, but more likely that it will move in the direction of the most magnetic rock in the general vicinity. read more
No, the Moon has no significant magnetic field like the Earth does. A compass would not be useful there. I think this is more interesting when we get to other compass types. Gyro compasses may or may not, depending on if the moon's rotation is enough to interact properly with a standard earth model. It does spin a lot slower than 1 day/rotation. read more