You will need a computer for the PADI AOW course. I recommend you get a simple one and become familiar with its operation BEFORE you take the course. I use a Mares Puck, relatively inexpensive and very simple to operate.
I agree, you should have a back-up plan in case your computer takes a dump. I wear my computer and compass on my right wrist, and my old fashioned depth gauge and watch on my left wrist. If my computer breaks, I use the 120 rule, with a 3 minute safety stop at 15 feet.