I think you'll be happy with the Mark III, Mike. Actually Canon's AF as implemented in the Mark III in combo with its 24L, 35L, & 85L primes is the only reason I'm sticking with Canon right now. The DR & Auto ISO implementation on the D800 would otherwise be enough to make me want to switch. Well, and native use of the 14-24
Note that for primes you'll want to microadjust based on your shooting habits. Since I tend to use the primes to shoot rather close subjects (that's more my style), I microadjust using a LensAlign at a distance of 25x focal length. I think manufacturers tend to suggest 50x focal length, as the best compromise. Certainly, using 25x focal length, infinity no longer focuses properly at wide apertures... it's a shame camera companies haven't implemented some sort of interpolation for microadjustment values based on subject distance (like they've done for focal length of zoom lenses). Maybe it's coming?
This does affect me in real-world shooting. For example, the other day I happened to be using my 85L in a non-standard way, shooting subjects more distance simply b/c of the way the event was set up. I noticed I got better results by resetting my microadjustment to 0 (which works best for distant subjects).
And then there are days where the microadjustment just seems to be off... haven't quite figured that one out yet... but luckily, it's not too often.
The good thing about the Mark III is that its precision
is good enough that usually you can tell if the lens if front focusing or back focusing. Not so much with my Mark II, which had so low precision that I just couldn't tell b/c focus was all over the place.
Now, I realize I'm placing high demands on the system b/c I'm shooting below f/2.0... but why else buy a prime if not for that almost 3D look of having a subject pop out from a blurred background? That's generally what I'm interested in when I'm using primes... not always, but most of the time.
The Mark III brings me closer to achieving that, without having to take 100 shots just to get 15 or 20 in focus at f/1.4.
The Nikon D800 also has very good (similar to Mark III) precision. But there are other issues with lenses that I won't get into here, but will hopefully write an article about soon. Furthermore, their focus points being totally miscalibrated with respect to each other is just something I don't wish to deal with. Luckily, my 5DIII focus points are pretty consistent (the leftmost one slightly backfocuses compared to the others, but its acceptable and nowhere near as drastic as what I've seen testing 4 different D800 bodies).