I would like to know if it was possible to get a pop up alert notification other than whatever browser you are using offers.  

Would be really nice if there was a clearly visible pop-up that overrides all windows and/or programs.  Even more nice if there was a button you could click on to answer the call with in the notification itself.   I know this is possible with other programs. 

Only if you install the Desktop Application version of the Client. Even then, it just gives you a "Show" button which, when clicked, brings up the interface so you can answer the interaction.

https://help.mypurecloud.com/articles/desktop-app/

Or you can install the PureCloud for Chrome app and extension:

https://help.mypurecloud.com/articles/install-purecloud-for-chrome/

That browser windows is "always-on-top" so it doesn't get hidden behind other applications...though that's not exactly what you're looking for, since you probably want PureCloud hidden until a call comes in, and just answer it from a notification without disturbing the rest of your work.