I hate to get all technical, but the chord is either a C7b5 or a C7#11. Part of the differentiation is functional, and part of it may be completely arbitrary, even if the latter is never admitted. In most practical situations, you're generally safer using the latter; although depending on how it's used in the context of the tune and the melody note, the former may be more appropriate.
But I've never encountered the #4 designation - that I can recall, at least.
The C7#11 implies that the 9th is also present in the chord (C E G Bb D F#)...the b5 implies that there is not a regular 5th present in the chord (so the scale would be C D E F Gb A Bb)...the #4 implies that the natural 5th is present (so the scale would be C D E F# G A Bb)...also, a "classically trained" string play would interpret, say, a Gb & a F# slightly differently pitch wise due to overtones & the relative "meaning" of each note to the key center..
But yes, you can have a C7#4 chord - C E F# G Bb. Unless you're really anal theoretically (and some people are...), it's more useful as a specific sound than anything else, but in theory (no pun intended) it does set up a different scale than a C7 #11 or a C7 b5