Well, you can plug any four-pole 3.5mm headphones or ear buds into a PS4 controller to get both system and voice chat audio that way, same way you can in most PCs. I believe the XB1 does something similar, but I'm not sure if that's just voice chat only, or system audio too.
I know some people don't like Turtle Beach, but I've been using their
PX22 for over two years and have no problem with it. On its own, out of the box, it will support PS3, PS4, PC, XB360, and anything that will take a four-pole 3.5mm jack ( phones, tablets, etc. ). With an
XB1 adapter, the headset will work there as well. Altogether you're looking at about $100 for everything. I'm sure other headsets are available, probably for similar prices, but this is one I use that I know will work well.