I have used it with a normal Apple headset, as included with iPhones, iPods, iPads - no problems. Most other headsets/earphones also have 3.5 mm 4-pin plugs, it is only important that a microphone is also included in the control unit.

On the Fritz site you will find an explanation of your question about the headset in the knowledge base.

I bought the Gigaset ZX400 to go with it. But be careful: Since the Gigaset has a 2.5 mm jack plug and the C4 a 3.5 mm jack, you have to buy an additional adapter.
Unfortunately, I have a very strong echo for the person I am talking to. Plugged into the mobile phone, however, it was OK. I.e. the C4 is the "bad guy". Maybe it's just me...