Apple - Is there any keyboard shortcut for entering El Capitan’s split view?
BetterTouchTool (starting with version 1.22, currently available as alpha version) let's you assign gestures or keyboard shortcuts to trigger the native split screen.
There doesn't seem to be any way to achieve this. I've trawled the web in search of an answer.
When I first moved to Mac from Linux, I picked SizeUp for this kind of window manipulation. It's the best tool I could find that has a full range of keyboard shortcuts. I'll be sticking with it for the foreseeable future. (It costs $12.99 but the free trial version doesn't expire)