While we do not offer a department-based feature or routing system where one chat box can be setup to route to different groups based on a visitor choosing, you can set up something close.  

On the Pro plan, you can create multiple chat boxes from app.purechat.com/websites. Each chat box can be customized separately and comes with a unique code snippet. Then on the same Websites page, click the user icon for each one to assign the desired user(s) to that particular chat box/website.  

A user can only get alerted of chats and see the chats they are assigned from the Websites page. A user can be assigned to multiple chat boxes and all chats come into the same dashboard. This is a common setup for companies that manage chat boxes for multiple companies in one account.