I don't know if this is a duplicate, but personally I feel like this one thing would be a good addition. Maybe make it switchable (so you can turn it off), but I don't think these things happen too often to be annoying. Right now when comments are moved, simply for their amount and not really that they are off-topic, that kills the conversation.
It often happens to me that someone answered me in comments, comments have been moved to chat and I don't see the response. I am not sure if it's possible to pertain all the pings when comments have been moved to chat. But it would be nice, of course, if that was implemented.