No, it isn't possible to have a fixed gear that shifts (at least not by a derailer mechanism). The reason is because the rear wheel can't coast freely. This means that the chain is turning CONSTANTLY when the rear wheel is moving, so derailing the chain would introduce a lot of weird stresses and would jam up the rear wheel (probably causing a crash), if the shift weren't incredibly fast. Moreover, you can't pedal backwards if you have a derailer because the derailer is not designed to experience stress in that direction. It will get bent out of shape if it is highly stressed in that way.

Waaay back in the day, Sturmey Archer made a 3-speed internally geared hub that gave you 3 speeds on a fixed gear. This was like 50 years ago. Maybe they'll make 'em again someday.

In any case, a fixed gear is a ton of fun. I built one for cheap and now ride it pretty much everywhere!

(I should mention that most fixed gears have flip flop hubs. This setup allows you to take the back wheel off, flip it around, and have a different gear on the other side. I don't use it, but the option is there!)