: ...if you were to peek out of a window of a spacecraft coasting at 0.9999c? Would your speed have any effects on how you viewed the stars outside the window?

of course...

RFL could answer this quite well. But I'll tell you best I know...

First, the stars would be close to infinitely blueshifted and you would be bombarded with x and gamma rays bigtime. Because the visible light that you observe traveling at non-relativistic speeds would now be very intense (bright) and much higher frequency due to time dilation effects.

This, however is not the only observation you would notice. I merely dealt with time dilation.

The distortion effect also would take place. Matter in your vector of motion would be length contracted, and your apparent distance to far away galaxies would be much closer in distance; compared to 'at rest' or non-relativistic speeds.