It would be interesting to compare the sharpness of the 1DX 1080 to the 1DC shot in 4K but downscaled to 1080. For those that want the highest IQ, the 1DC could compete with HD cameras if the extra PP conversion is not too much of a headache.
The 1DC is better for sure, here is a short that was shot exactly the way you described:
Now I have uploaded hundreds of videos to Vimeo shot on DSLR's, and Vimeo's encoder sucks, it creates artifacts and changes the color slightly. But I think even the Vimeo footage from the 1DC looks better than the footage straight out of the 5DIII.