And thereby totally removing the point of engaging in Social Networking. Plus, the second you post something you'll give the game away to those on your friends list who actually know you, triggering all the questions of "[name] why did you change your name?" Plus, the only people who'd want to be anonymous on Facebook are haters, trolls and stalkers.
How about set up privacy settings that restrict total strangers from viewing and just be done with it? Or how about don't post anything ridiculous to start with?