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I can answer my own questions instantaneously! That's Awesome! The problem is, a lot of the time, when I answer my own question, if it doesn't get any comments or a competing answer, I can... well, I can kind of forget about it. Often, I don't even realize that I haven't accepted my own answer until I see my accept rate starting to dwindle and start digging through old questions in search of a fix.
I don't think doing away with the 48 hour waiting period before accepting a self answer is a good idea. It allows a chance for someone else to see the question as 'unsolved' for a few days, and sometimes, a competing answer can be better than the one I'd written myself. The waiting period helps to encourage those competing answers to be contributed, as answering a question with an Accepted Answer already can be intimidating and feel pointless.
Instead, how about reminding me, say... a week after asking/answering a question using this shiny new button that I haven't accepted my own answer yet? A simple inbox alert similar to the Badge Award/Bounty Expiration notifications seems appropriate here, and it would help to encourage me to check in on my 'solved problem'.