The problem with trying to make this as realistic as possible (i.e. developing something akin to Doppler shift) is that it makes no sense to do so if you're just going to leave out other byproducts of traveling at/near the speed of light. For example, relativity states that as you get closer to the speed of light, your mass increases towards infinity, which is why seconds pass by for you and a much larger amount of time passes for those not going that fast. Someone else here in the forums asked something similar, and when I brought this up, they mentioned an Alcubierre drive (or something similar). The issue with that, though, is that you wouldn't (or shouldn't, anyway, although I could be wrong about this) experience the Doppler shift, as you technically aren't traveling faster than the speed of light - you're warping spacetime around you.