I switched from AIS to DTAC in 2008, quite a few years ago now and I've been on the same post-paid contract (and still have the same number) ever since. At the time all I needed was my passport and address if I remember correctly. Initially they give you a very small monthly credit limit, but I now have a 10,000 Baht limit and I think they'd give me more if I asked for it. As an aside, number porting has been available in Thailand for a few years now i.e. you don't have to give up your number when you switch providers.
I believe it is pretty simple with DTAC too. I recently registered my DTAC prepaid SIM. They just took a photo of my passport details, did something on my phone and on a laptop, and it was finished. I got an SMS saying I was registered. Deadline for registration is end of July.
I've been a 1-2-Go pay as you go phone user in Thailand for over ten years. However, changes to my contract mean I no longer have an Indonesian SIM from the company, so it's cheaper for International roaming to get a post-paid phone contract rather than continue pre-paid. I wasn't too sure how difficult they were going to make this having heard of requests for work permits and/or 'yellow books' so I called into the AIS service centre at Hat Yai Central Festival today with some trepidation. Indeed after I explained what I wanted the young lady asked to see my passport and work permit, but after explaining I was here on a non-O marriage visa and worked in Indonesia she was happy enough to settle with my passport, my bank book and my wife's ID card. Forms were filled in photocopies signed, a few taps on the keyboard and hey-presto I was switched from pre-paid to post-paid. All remarkably painless.