The star, Alpha Centauri, is 4.3 light years away: could we watch an Alpha Centurian TV show that started 4 years ago? Yes, but Alpha Centauri is moving towards the earth at 46,000 miles an hour, so the TV show would be fast forwarding, just like if Alpha Centauri was moving away from the earth, the show would be playing in slow motion. However, if a spaceship from Earth was passing Alpha Centuri 4 light years away, it could watch the show at the correct speed. What if a spaceship from Earth was traveling away from Earth at lightspeed, and it was 4 light years away, could it watch an Earthling TV show that started 4 years ago? No, because the signal wouldn't be able to catch up.

It get's weirder. What if a spaceship was traveling at lightspeed and the astronaut was trying to watch a movie playing inside the ship? If the monitor was facing forward, the light from the screen would never change because the astronaut was traveling along with it, and if the screen were facing towards the back, the astronaut wouldn't see anything because the light would skip right past him. Now consider that everything else is moving too: Earth, Galaxies, the universe; it's remarkable you can watch Alpha Centurian TV at all.