If you're considering an Xbox One, you should know there's no official adaptor to connect existing Xbox 360 (or any other) headsets to it. All isn't lost though. A headset comes with the console, and you can hack it into an adaptor that works with your favourite headset. It's surprisingly easy.
Instructables users octanechicken and DragonballZ Deep both had similar ideas, and both disassembled the Xbox One's flimsy little headset and turned them into adapters that will work with any other headset you choose to use. The former ended his with a 2.5mm female plug, the latter ended his with a 2.5mm male plug. You can choose the one (links below) that you prefer.
Both mods require you to crack open the headset that shipped with the Xbox One and do a little rewiring, so say goodbye to it -- it's probably no big loss when compared to what you'll get in the end. You'll need a soldering iron, but ultimately what you'll need to do is either cut or desolder the headphone cable, separate the individual wires, and then re-splice them to the adaptor you want to use. Each walkthrough explains exactly how to do this. Once you're finished, you'll have an adaptor you can plug in to your current headset -- or just about any headset on the market -- to use with your Xbox One that will be far more comfortable and sound better than the one included in the package.
If I had to pick one, I'd choose octanechicken's walkthrough, since it will work with any headphones you plug directly into it via 2.5mm plug, while DragonballZ Deep's walkthrough ends with that male plug that you'll have to plug directly into your headphones, assuming you have headphones that can use a patch cable. Either way, hit the links below, see which one works for your preferred headset, and give it a try.
Xbox One 2.5mm Headset Adaptor [Instructables] Xbox 360 to Xbox One Headset DIY Conversion (Turtle Beach) [Instructables]