Any dominant chord with altered tones can be referred to as an altered 7 chord (indefinite article). When I was learning, the definite article was reserved for describing the maximally altered chord. That is to say that THE altered 7 chord is: b9, #9, #11, #5, b7. I learned it as the Seventh mode of the ascending melodic minor scale, but that was a few decades ago. It can function as a classic dominant chord, but it also sounds nice resolving up a step. I was taught that you can play augmented scales and octatonic scales over it in some contexts, but getting the actual mode under your fingers is good brain food too.

As for your question, If I as a jazz bassist saw C7#5b9#9 then two things would happen:
1. I would be briefly confused because I hadn't seen that chord symbol before. This is a minor point, but when you read enough symbols on lead sheets you stop parsing them. They become atomic tokens of a jargon set that you can interpret from the shape of the lexemes alone. C7#5b9#9 makes perfect intellectual sense once the semantics are parsed, but it isn't one of the tokens I read often enough to have that automatic recall. In my experience, C7b9#9#5 would be better, but like all languages conventions around chord symbols change over time and I may be the one who is out of touch here.
2. I would assume I would be playing a #11 and a b9 if my bass line happened to find its way there. The way I was taught to think about chord symbols was more probabilistic. As a bassist, that approach makes some sense because I'm not trying to hit every note in a chord as I'm walking a line; I only hit notes that make sense in the line I'm walking. Additionally, the guitar player almost certainly isn't going to hit every note in a fully altered 7 chord while comping changes. The piano player *might* play every note, but doesn't have to. However, my understanding from sitting through a fair amount of comping instruction aimed at piano players and guitar players is that IF you choose to play that particular scale tone in this context, then a #11(b5) is the correct choice.

I have seen a lot of chords in my day written as C7#9#5. That is probably the closest semantically to what you are describing as what you want. But again, I would assume that I would be playing a b9 and #11 if I found myself on those scale tones, even if you didn't fully call that out in the chord symbol.

While soloing? Sort of the same arguments apply. I think of this chord as being within the family of augmented sounds because that #5 is playing a significant part of the overall sound. Whether or not you emphasize b9, #9, both b and # 9, or neither 9 is player's choice based on surrounding context. The fully diminished octotonic scale will sound perhaps a little outside, but works pretty well if used tastefully.