Sony has confirmed that the PlayStation 3 Wireless Headsets, including the Pulse headset, are compatible with the PlayStation 4. Players will not require an additional adapter or any such device to make them work. This is good news for players who did not want to switch to another headset in the future. If your current PS3 wireless headset is working, hold onto it.
This information was given by a Sony representative at Comic Con. The PlayStation 4 will contain a headset, but that is for online chat; not in-game sounds.
On the other hand, the wired Xbox 360 headsets are not compatible with the Xbox One. Microsoft claims that this is because of differences in the controller ports. However, an adapter might be available in the future which will allow the old headsets to be used with the new console.