Money and fame don't always go hand and hand. While anesthesiologists rake in the big bucks, they rarely rake in the press...as long as they're doing their job properly, that is. Being famous in this gig usually ends up closer to being infamous—like, when someone fakes the results of their research papers or become an evil doctor who injects patients with illegal drug cocktails.
Or maybe some medical error caused the death of a famous rich patient, say a Michael Jackson-type, causing the doctor's name to be splashed all over the front pages of tabloids throughout the world.
But you wouldn't want that type of fame. You'd want to settle for what the rest of your colleagues strive to achieve: a good reputation within the profession as you work hard to banish pain among the sick and benighted of the world.