Yeah, that is just the pattern of the noise caused by severe underexposure, usually made much worse by attempting to brighten the scene.
Sometimes if you have an area of deep shadow, in an otherwise correctly exposed scene, it is best to just leave that area dark, and not try and pull detail out of it as it will bring out pattern noise like this, that otherwise you wouldn't see.
Apparently, the 5DIII also shows this tendency, but I have never personally seen it on mine as I don't try and pull up such underexposure. If I really need shadow detail, I take a second shot exposed for the shadows and blend them (probably not much help if you are shooting video though).
All you can do is try not to underexpose so heavily, if you have a problem shadow area, you need to add light to it, when you take the shot.