All Canon extenders mount on the Sigma 70-200 2.8 OS. I used to have 4 different 2x TCs, including the expensive Canon 2x III. Sold it eventually when I got my Sigma 2x APO EX DG. Sharpness in the center was close, actually favored the Sigma a bit. Corners: Canon won by a bit. What the Canon does better is controlling CA. I ended up selling the Canon and freeing up some funds this way.
However, the question is what are you planning to shoot? If mostly birds, I'd go longer, such as Sigma's 150-500 OS, or perhaps the Canon 100-400L IS. Yes, it is only 400mm but a good copy is nice and sharp and contrasty. The Canon 400m f/5.6L is good too, although no IS.
If you think that you could use a fast 70-200 lens in many applications, it is worth considering the Sigma 70-200 2.8 OS or (if budget allows) the Canon 70-200 2.8 IS II.
I'd say a Sigma 2x TC works pretty much equally well on the Sigma lens, but if you go the Canon route, I'd get the Canon 2x III.
The Sigma 70-200 2.8 OS is decent with the 2x Sigma TC, I usually avoid wide open (f/5.6) it is much nicer at 6.3 and up. Still, even an average copy of a 100-400L would beat it in IQ and especially AF speed.
I also own the Sigma 120-300 2.8 OS which is a great lens. Sharp, and fast, and works great with TCs. But it's quite heavy. So quite often I actually take the Sigma 70-200 OS + a couple TCs with my for a "nature oriented" walkaround. When I know that I'll be sitting in a blind or other stationery places, I take my 120-300 OS and tripod/monopod.
but again the bottom line is, do you plan on only shooting birds, or do you also shoot events, portraits, sports, especially in lower light environment? If yes, then a good 70-200 2.8 OS/IS + TC would be a good, flexible solution.
Here are a couple shots taken with my Sigma 70-200 OS + Sigma 2x TC (cropped somewhat and processed):
(Note: the last one was taken using an older, non-reporting TC, hence the EXIF info stating 200mm)

IMAGE LINK: http://www.flickr.com …/29108710@N04/6982833742/
Great Egret with fish 6
by
gabebalazs
, on Flickr
IMAGE LINK: http://www.flickr.com …/29108710@N04/6941296912/
Great Horned Owl Babies 13
by
gabebalazs
, on Flickr
IMAGE LINK: http://www.flickr.com …/29108710@N04/6193055305/
Egret with Sigma 70-200 OS and 2x TC
by
gabebalazs
, on Flickr