thing to think of doing, and I wouldn't dream of it for a moment if I was you.
The fewer people who know when you're online the better, and even then they should be friends, or business associates. A computer broadcasting the fact that it's sitting there, waiting for contact, is a hacker's idea of heaven. We all broadcast the fact anyway, every time we connect, but there's no point in shouting it aloud.
If you want to have a (more or less) real-time conversation with site visitors, run a forum on the site and let them have free access. Whenever you're online you can simply open your forum and keep it in your taskbar - checking it from time to time.