In order to redshift you have to be moving relatively extremely fast toward another object, and if you're moving that fast, you can't see a computer screen as you fly past it, your eyes wouldn't focus on it in between when it was close enough to be in your sight and before it was out of your sight (aka behind you).
Nope. No noticeable redshifted BSoDs.
(Though I admit, a very clever idea. I just felt mean.)