Observed speed of a receding light source Let’s say there’s a planet 4 light years away from Earth and we send a rocket ship towards that planet at 99.9% light speed. We stay behind on Earth and watch the rocket ship travel towards the other planet.
Eventually we should be able to see our rocket ship reach it’s destination. How much time will have elapsed for us until we see that occur?
My intuition would say about 4 years. But I also know that when we observe such a far-away planet, we are ”seeing it as it was 4 years ago”. Well 4 years ago the rocket was still on Earth, so how can I be seeing it landing on the planet now?
Something has to give, but what? Will it appear as if the trip took 8 years to complete?
 A: 
Will it appear as if the trip took 8 years to complete?

Yes.
A: It takes the spaceship 4 years to get there, and then it takes light from the spaceship getting there 4 years to get back, i.e. 8 years for an observer on Earth to see it land on the planet.
A: It all depends on what the meaning of the word "appear" is.
In about eight years, you'll see the ship land, and you'll say "Ah.  I see the ship landing at a place four light years away, so it must have landed four years ago".
Does "appear" refer to what you see, or to the meaning you attribute to what you see after you've made the necessary corrections?  If the first, the ship appears to land in eight years.   If the second, it appears to land in four.
An analogy:  Suppose I stand really really far away from you.  Would you say that I "appear" to have shrunk to half my height?  By one meaning of the word "appear", the answer is clearly yes.  By another, no, I don't appear to have shrunk, I just appear to be really far away, but to still be six feet tall.
