No, not another site. In the long run, with people participating in multiple SE sites, SO, SF, and SU, there will be all the meta sites to participate in plus all the chat sites. This is too much.

stackoverflow.com meta.stackoverflow.com chat.stackoverflow.com serverfault.com meta.serverfault.com chat.serverfault.com superuser.com meta.superuser.com chat.superuser.com unicornbreeding.com meta.unicornbreeding.com chat.unicornbreeding.com 

I'd rather introduce a special type of question into Meta that has Chat format, or try relaxing Meta rules even more than they are already (not sure whether that won't swamp the place with crap, though).

Too many sites.

I'd rather introduce a special type of question into Meta that has chat format, or try to find other ways to integrate this stuff with Meta without turning it into a crapswamp.

If this is at all necessary, Meta is the place for it IMO. The "Work / Home" analogy doesn't really apply because SO is not really "work", and Meta is not "home".