I only have a mouse and a keyboard right now, nothing else. Of course it's good to have MIDIcontrollers it can make the workflow easier and more entertaining, but u don't need any to get started. I had a cheap midi keyboard and some chinise junk pushpads, but as u can guess, they're all dead by now. I'm looking forward to buy new ones, but those decent ones that I need are relatively expensive.

So in a nut shell: no, u don't have to buy anything to use ableton, you can even perform live action without any gadget if u're that dedicated.