Yeah, I frequently use that extra bass G in the open position C chord (or even extra heavy barre chords)...especially as an ending chord to get an extra full sounding resolution.
But let me also throw out there that you can just as easily add an extra G on the high E string instead (still fretted with pinky). In general, this is used for more melodic phrases. Either way, it's still a complete C chord, as E is still fretted with the D string 2nd fret...it's just another way to "voice" it, as explained above.