What happens when you generalize Probability Theory to allow Minus Signs ...

Over at Shtetl-Optimized, Scott Aaronson has a nice introduction to the field of quantum computing disguised as a series of answers to questions he was asked on his recent job interview tour.  I particularly like the following characterization of quantum computing as:

what happens if we generalize probability theory to allow minus signs, and base it on the L₂ norm instead of the L₁ norm

I now feel I know enough to be able to talk authoritatively on the subject over coffee at work.