Is there any app or hack which can split a monitor into multiple screens? So for instance, a 1920x1080 display could be treated by the operating system as though there were two 960x1080 screens connected instead.
There is No: True Multi-Monitor Support for a Single Display
There is no way to get the OS to treat a single display as more than one display.
You Can: Fill Defined Sections of your Screen with an App
Adding my vote for Moom! Excellent - as simple or complex as you need it to be, and keeps out of the way. http://manytricks.com/moom/