Not that I'm against it in any way, but it's not really a necessity. There is tons of IRC software that can be used to access IRC. Alt-tabbing can't be considered "leaving the site". Not to mention there's IRC software such as Chatzilla(firefox) which is basically a part of your browser.
What you are suggesting sounds more like a webchat, something that would be on the front page. Having such a chat means you cannot leave that page at all, making it moot as you cannot "use" the forum. (Unless you have two tabs of the website, but then it's the same as having one tab for the forum and some IRC software to tab between.)
You could go the awesome and expensive way - buy another monitor. (it really is awesome having that, unless laptop ^^ )