So it turns out you can fool your PC into accepting the new Xbox One controller, by installing drivers and software that convinces your computer that it's actually been connected to an old Xbox 360 controller.
Lucas Assis, a modder, posted a tutorial on Youtube, and by the looks of it it's fairly straightforward (if you know your way around Windows).
He developed the workaround in just under a day, and while he says "it works great", he also notes that "it's not really that easy to install". Check it out for yourself if you're interested in hooking your Xbox One pad to you PC.