While it's tempting to think that with the Magic Lantern EOS raw hack out in the open that normal, compressed video will never be good enough again, that's taking far too strict a line on it. The fact is that the conventional video output from cameras like the EOS 5D MK III is very good indeed with the exception of certain well-known issues like moire and rolling shutter
The latest (official!) firmware for the 5D mk III brings the ability to output "clean" video over HDMI. This means that instead of seeing "viewfinder" information in your output, what you get is an HD signal that is good enough to record. And because it is sent through the HDMI port before it is compressed, it is at least possible that you'll get a better recording if you capture it externally with an external recorder.
We've mentioned the combination of the 5D mk III and the Atomos Ninja 2 here before; but we've found a much more comprehensive video for you which is intended not to promote the Ninja so much as to help you understand how this type of set-up works, and to illustrate that there is life outside of a raw workflow.
Here's Stefan Czech with his forensic analysis of the 5D mk III/external recorder combination.