I strongly believe that a user, whose comment is deleted by a moderator should be notified about it.
Many moderators ask the user to remove their comments however some just remove them without informing the user at all.
The ethos in the MSE answer linked in the comments,
Comments are third-class citizens in our system by design; they are little more than post-it notes on the units of real work, questions and answers. As such they are afforded little protection and the burden of proof is on the comment to be useful and constructive.
remains the core component of Stack Exchange's attitude towards comments on this platform.
As such, there is basically no chance that SE will implement this notification feature.
If you want something a bit more definitive, try the answer (by a former SE dev) in Notify user when their comment is removed:
There are no comment removal notifications, nor will there be.
Why? Because comments are second class. We don't see them as important - if you think a comment is important enough, it should be rolled into the post it is commenting on.
Removed comments basically fall into three somewhat-overlapping categories:
The comment was part of a discussion which led to improvement in the post it was attached to. Since the improvement has been achieved, the comment can be removed.
The comment didn't contribute to improving the post it was attached to, but was instead an answer, a part of a side discussion, or a joke.
The comment was actively harmful in some way --- rude or spammy.
There's not any reason to notify folks whose comments fall into the first category; such notifications would just be noise.
Notifying folks whose comments fall into the third category can be actively harmful. The main reason to remove rude comments is to de-escalate tensions among people who are getting fighty with each other. Someone who is already fighty is likely to respond to a removed-comment notification by lashing out, or by repeating the troublesome remark, or the like. It's much better if those comments disappear silently: if their absence is noticed, it's likely to be later, after tempers are cooler.
I forget how notifications work if a side discussion is moved to chat. I don't think there are any, apart from the moderator comment that gets generated when the comments are moved. I'm totally okay with that, for the reasons discussed in Emilio's answer.