Also interested in this functionality.
IMO, available chat iterations are either overboard or not placed conveniently or persistently throughout forum pages. This causes a circumstance where you need to have multiple windows open to monitor "chat" and/or browse the forum at the same time.
Alerts help but a simple user selectable toggle to allow PMs to pop up in real time would be sweet. No separate 'chat window' or specific forum page would be required.
Simple is good.