End-User New Message Notification

I am wondering if anyone has found a way to add some sort of indicator to the end-user that they have a new message when the widget is minimized? Although the asynchronous conversations through email are handy, our business would prefer not to have sales lost because a customer doesn't realize we've responded to their question before they've checked their email, and unfortunately a significant portion of our customers are not technically savvy to the point of understanding they would need to open the widget to see if there has been a response.

Would love to see this as a product enhancement.