Even a standard wired headset for smartphones/cellphones should work fine with the PS4 so long as it uses a 3.5mm connection. I haven't come across a wireless option yet, not like the bluetooth wireless headsets that work with the PS3. I have a feeling they are coming but, at this point, Sony wants you to plug your headset into the DS4.
Depending on exactly what you want, there are several options available. The Apple Iphone 4/5 earbuds with built in mic work well. And Walmart sells a few different headsets for the PS4 in the $20-&100 range
The iPhone earbuds do work for audio on the ps4 through the DS4, but when using it as a mic, it causes a very loud and annoying hum noise that everybody will be able to hear. I wouldn't recommend those.
to the OP, you could find just about any pc based USB headset and use it if you don't mind a whire running across the floor. You can get some fairly cheap. I personally however, picked up the pulse Elite headset on sale, it was just over $100 and I'm very happy with it.
Looking for ps4 compatible headsets under 100$
Make yourself one using a headset you already have. Instructions here.
Uh, spend another dollar and use a jack.
Any stereo mic input will work, but, I snagged something similar to one of these at a Staples for $11, and used a chip clip to affix it to the headphone cable I had, via the ear hook:
Okay, with me so far? Since the headset included the mobile-to-pc adapter, the two parts combined cost me around $16. This leaves us $83 to mess with for a stereo headset (for the record, I use these, and have been insanely happy with their performance for the last twelve years - yes, twelve). If you want surround sound, you can get a first-generation DSS for $30:
This leaves us with around $50, give or take:
And, you even have enough left over to get a headphone amp, if the signal isn't strong enough:
I realize that's a whole lotta wires, but, it all works, and leaves it up to you what you want to upgrade, and when. of course, if you already have a great pair of headphones, and/or surround sound doesn't matter to you, you are in great shape.