The video game Mass Effect goes into great detail on how the Doppler Effect changes what you see when traveling faster than light. Wavelengths of light are stretched or compressed, altering the color of stars, even shifting your vision into infrared or ultraviolet. But the Doppler Effect is not the only thing that distorts what you see at such ridiculous speeds.
I recently found this video on Astronomy Picture of the Day which demonstrates what you would see if you traveled near the speed of light, taking into account the Doppler Effect, relativistic aberration, and the so-called headlight effect. The world becomes increasingly distorted from that point of view, mainly because you are no longer seeing the world as it is but as it was.
For the purposes of the video, light travels at one meter per second (as opposed to 300 million meters per second). Thanks to the Theory of Relativity, it’s not the speed of light that really matters here but your speed relative to it, so the demonstration is still accurate.
Traveling faster than light is not possible except in science fiction (and maybe for neutrinos) and in the video the car or spaceship or whatever never travels faster than light. But we SF fans and SF writers can imagine, based on this video, how much stranger the universe would really look at warp 9.