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.)
Actually the speed of light is a universal constant, it does not depend on the container that holds it. This means unlike sound which would travel through a moving objects cabin as if it was stationary, light would appear to slow down inside a rapidly moving spacecraft (when fired from rear to front). Because of this you can have a screen in a fast moving craft that is next to you and you will see the redshift occur. The rest of your cocpit would shift colors as well.
What this means is as you go faster in a space ship facing toward your screen which for some reason is situated at the rear of the craft, it would appear to turn red. If it were at the front firing back it would turn purple/ultraviolet and beyond.