In reply to TomAlford:
I use a Canon 700d with a 10-18mm lens.
For me, it's a good compromise between quality and practicality - it's not as heavy or bulky as the 5d/6d/7d or even the mid-range DSLRs (60d etc.) The quality of photos is not as good as those taken with a 6d + L-Series glass but that much is obvious. The whole setup cost me about £650 a few years back so should be cheaper now. A fraction of the price of a beefier camera.
I was concerned about the cold and moisture (cheaper Canon models are not weathersealed like the top-end models) but I've consistently used the camera for Scottish winter (i.e. cold and wet), Norwegian ice (i.e. extremely cold and extremely wet) and alpine winter (i.e. "why are we out here again" levels of cold). It's always been totally fine. Have dropped it in deep snow a number of times and never had issues with moisture. Keeping batteries in your chest pocket keeps them warm and makes them last longer.
I'm surprised no-one has mentioned this yet, but if you want to take good photos in the mountains, I consider a super wide-angle lens essential. I used the kit lens (18-55m) that came with my camera for a long while, but my photos improved significantly when I bought a wide angle. The 10-18mm is one of the cheapest lenses Canon make but it's stood up to a lot of my abuse.
Another important consideration is a carrying setup that will allow you to actually use the camera while climbing. Carrying the camera in the top of your sack and promising yourself you'll be disciplined and get it out on belays just never works (in my experience). I use a modified LowePro pouch mounted on a chalk bag belt and clipped to my harness with a screwgate. The camera inside is then attached via a cord lanyard so I can't drop it.
Hope this helps!