I did geological mapping with a sighting Brunton compass (about $500 in 1978), and navigated 8 wihiteouts by map and compass alone. I always carry backup map and compass with me.
And I find the GPS is superior and vastly more convenient. I have this vivid memory from a whiteout in 1983, with blowing snow and 50 mph winds. I got the map, carefully folded it into a small rectangle inside my pack, took it out in the light, and had the wind mercilessly rip it to shreds. I have navigated two whiteouts by GPS since, and it seems so vastly safer and easier.
I carry the gps very high on my body, so I get better signals in canyons... but if you are in a very deep canyon, you don't really have a lot of choices where to go, do you?http://hwstock.org/RainGunJuniper13/html/DSCN2826.htm
I often get asked the question "but what do you do when the batteries run out?" Well, I generally don't start out unless the unit has freshly-charged 2900 mAh NiMH; and when it does run low, I take out the 2 Li metal batteries that I always carry in my pack, and replace the old batteries. The need for field replacement has occurred 4 times in over 800 trips with the GPS. Li metal has a long shelf life and operates well in cold.
Sometimes when I'm in the outdoors, a person will start to wax poetic about the superiority of the compass. I ask that person a simple question: do you know what the declination is here? Now, in this year? I've never had anyone able to answer that question correctly. Most people simply don't know how to use a compass, except to get a vague feeling of where north is. If you know how to use a compass well, good on ya; you are in a minority.