I use CyanogenMod on my S3 and it works pretty well. You'd probably want a 4.4 rom since that's KitKat, and most new developments are only happening on that version right now. For instance, if you have a Chromecast you can use MirrorEnabler to cast your screen to the TV, but only if you're on KitKat or higher.
One thing to note if you decide to go with a 4.4 rom: UPDATE YOUR RECOVERY SOFTWARE FIRST. Whether that's ClockWorkMod or TeamWin Recovery Project, they need to be updated to a certain level or else a KitKit rom installation will brick the phone. Learned that one from experience, heh. I personally used the "TWRP manager" app to update it the most official way I could find.