I think I understand. How about you tell your friend this:
coffeebean wrote:
I can always set my status to Away, but it's very rare that I only focus on chatting (everyone else I know also plays games, reads, listens to music, etc, while online) and inevitably that's probably going to become a problem, too, since I'd also be chatting to other people.
Maybe your friend doesn't know that you are doing other things while chatting, so it seems like you are just ignoring when you don't respond right away.
I would be kind of like your friend in this situation. I expect to have a person's full attention when we are talking, and I don't even understand how people can do the multitask-chat thing. If your friend is like that, chat may not work for the both of you. Or you could set aside a time when you are only chatting with that one person.