This question has engendered a good few comments, ranging from the merely patronising to the frankly abusive.
Do we have a policy/protocol on such "conversations"?
Flag the comments.
On every comment, you have the ability to flag it as "abusive" or "rude". This will bring it to the attention of the moderators of the site, who can then delete the comments and take further action (such as warning users) if necessary. If a comment accrues enough pending flags, it will also automatically delete itself.
If a conversation has turned toxic, it's also a good idea to disengage. Leave it alone, flag it, and wait for it to be taken care of. There's no need to continue leaving comments once it's clear that things are not going to turn out well.
As a followup to Mith's answer and the ensuing comments, it looks like the mod response to the flags (not me, I was frolicking through the Land of Nod at the time), was to delete some of the worse examples and then move the whole rest of the mess to chat.
That's about what you can expect for a flagging in this situation. In theory of course, chat is supposed to have the exact same niceness standards as the rest of the site. In practice though, we are a hair looser there (but still don't condone outright abuse, of course) because moving things there has a distinct tendency to cool down the discussion.
I personally believe this is because its not as public, so egos aren't quite as involved. Also, there's a barrier to participation that casual readers aren't very likely to cross. Thirdly, the more interactive (and real-time) nature of the chat just makes everything said much more like a personal discussion, which seems to tamper down all but the worst actors. So what I tend to see happen in chats is either the whole thing gets dropped, or the two main parties eventually come to some kind of an agreement (even if its an agreement to disagree).
Don't think that every move to chat is being put in the penalty box though. Sometimes we will also use it for a really interesting discussion that it seems a shame not to save (remember that comments are our barn cats).
Followup to the followup - In this case the user appears to have been one of those not "tampered down" by a discussion being moved to chat. Chat comments were flagged and deleted (which comes with an automatic chat suspension, as well as getting site-wide moderator attention). I generally don't know that its appropriate to publicly discuss moderator actions wrt. specific users, so I will end the story there.