They are both awesome bodies with fantastic IQs. On Canon dedicated forums (@ POTN) you are bound to have more answers saying get the Canon. But the truth is they are very close to each other in performance. If you are not already too heavily invested in Canon glass the Nikon D600 is certainly a viable option. And this "oil in sensor" is totally blown of proportions here..
Some basic advantages for D600:
-More AF points (better AF tracking, in 6D you are still basicly stuck on the center AF point for most reliable focus)
-2 EV stops better dynamic range (in landscaping photography you'll appreciate this)
- 4 more megapixels
- built-in flash
- better low ISO perfomance (e.g. ability to pull out shadows without that horrible banding)
Some basic advantages for 6D
- better high ISO perfomance (it becomes more especially visible above ISO 6400, but you have to ask your self: do I really shoot with such high ISO numbers?)
- If you are already invested in Canon glass, in my opinion it's not worth changing them for Nikon
To sum up: they are both awesome bodies. Just go to some store, see how they feel on your hand. And make your decision based on that feeling
I did a lot of research before deciding to go with the Nikon D600 largely for some of the reasons posted above. However, my D600 arrived and I tested it right away and found about 15 spots on the sensor. I didn't want to deal with sending it in or needing to clean the sensor time and again, etc. So, I sent it back and decided to rent a Canon 6D and the 50/1.2L lens. I had it for a week on a trip to New Orleans and returned it and bought a 6D and the 50L right after that. If I knew that I'd like it so much back when I was doing my research then I would have probably gone with the 6D over the D600. I often shoot available light and the 6D's center focusing point is unrivaled in how well it can focus in even the darkest light. I love it.