May be It would be better to use desktop app for this purpose. Also we are working on React version. It can be integrated to any part of the page. 

About default version: for me it is ok to use messenger in another tab when I checking other page of the site.

If you like to open messenger in new window you may add something like this window.open('page.php?i=messenger'); in studio -> navigation -> account notifications to onClick filed and also make empty URL option for the same messenger menu item.