No. I completely disagree.
I don't think anyone should ask a question to which you think you already know and have the best answer.
If you are trying to get new ideas and think one particular way of thinking is the best, then you should include it in your question and ask whether that is best. That is okay.
The one place I think it is okay to answer your own question is when you really don't know the best answer, and the answers given are not quite it either, but they lead you to thinking or further web research and then you discover the best answer. Then you can include it and state so.
But preferable to that, you should make sure you wait long enough, say a few weeks or even a month, for someone else to come up with the same answer you discovered. Then you can and should give them credit for the answer. And you can document any subtleties in comments to that answer or by editing your question with a followup.
And waiting the few weeks or month without accepting an answer will result in more answers, giving you more ideas, and maybe a better solution.
Followup: I see that Stack Exchange is not only allowing answering your own question. They're actually promoting it!
Click "Ask Question" and on the page just below the "Post Your Question" button, you'll see a line that says:
Answer your own question – share your knowledge, Q&A-style
At the beginning of the line is a checkmark. If you select it, up pops a text box allowing you to immediately supply your answer.
The link on that line takes you to a blog post by our supreme leader Jeff Atwood that says It’s OK to Ask and Answer Your Own Questions.
Well, rather than the one exception I stated above, I still very much disagree with doing this and think that promoting this when we ask the question is doubly bad. I still think one should only answer their own question after others have had a chance to do so - and especially should not do it immediately when they ask it.
p.s. I've added most of this answer as a comment on that blog post.
I've also added this to: "Can I answer my own questions, even those where I knew the answer before asking?" on Meta Stack Overflow.