A good compass need not weigh more than 20 grams (3/4 oz), but there are even lighter variants if you follow trails and do not need to plot a course.

This is not the lightest of compasses, it weighs just over 30 grams (1,1 oz). But it works very well and the mirror is great when you want to comb your hair.

This compass is slightly smaller and still holds a mirror. It has the added advantage of a safety pin (look closely). This pin can attach the compass to your clothing at chest level, thereby enabling you to have your hands free (for walking poles maybe) while at the same navigating and constantly keeping an eye on your compass. Particulary useful in fog, when you might need to use the compass for long periods of time in order to stay on course.

Conclusions and new findings: In the mountains you can usually navigate quite well with only a map, but I would never go without a compass. You never know when poor visibility might hit, or when you need to navigate cross country in bush or forests. There are lighter varieties of compasses than the ones I use, but I find that the 10-15 grams to be saved that way are of no importance. Maybe even ridicolous. At least to me.