You want to try BuddyPress Desktop Notification? No problem! Just register two accounts and send notifications from one account to the other!

For your interest: Unread messages and unread friendship requests will be send with every page load. The ajax request for status updates considers two things: 1. has this update been made after the page load and 2. has the current user been notified about this update? So if you update one account and then switch to the other, your status update won’t show up, since it is too old.