GUEST, I use CGCGCE. Many Celtic players use C modal in which the first string is tuned down a tone, thus CGCGCD.
I think that your tuning of the fourth string to E, although it is still an open C tuning, would cause the tuning to lose much of its versatility, although, no doubt, there are effects possible with it which couldn't be obtained any other way.
As a general rule, with any open tuning you want to keep the major third out of the bass strings, so I would suggest it is almost mandatory to tune your sixth string down to C. Also, having two thirds sounding in the open chord isn't a good idea.
The unique beauty of the CGCGCE tuning is that it keeps the major third right up on the first string, out of the way.