Ive been playing Tales of Xillia 2, FF XIII-3 and other assorted games with my DS4 without any issue.
Not sure what "Compatibility" issues arise, but you will HAVE to have a DS3 with you at all times that is charged.
The controller is seen as a "3rd party" controller, so the PlayStation button does not work (for quitting games, etc), though it will turn on the console.
I think it is a vastly superior controller to the DS3.