*** A new used door is going to cost you a lot of money. Installing a door can be a lot of work, depending on what needs to be switched over from the old door. If it was me, I would:
1. Go back to the shop that painted it and tell them it's unacceptable and ask them to re-paint it. If the door was repainted as a result of an accident and the claim paid by insurance, contact the insurance company and tell them the repair is poor and unacceptable and the value of the car is diminished. If your insurance company is like most today, they probably won't care, but it may help.
2. Just fork out the cash to have it repainted by someone who knows what they're doing. It's not too bad since its only the door itself. A good body shop can fix that.
All the compound and polishing in the world won't fix that. I had a similar problem with my old car that had the rocker panel re-painted titanium silver, and it didn't match. I tried a lot of stuff to fix it, nothing will. The problem is with the basecoat, and polishing doesn't effect that. I have to agree with the others though, it looks pretty bad. The fact that you can only notice from certain angle doesn't / shouldnt improve your opinion of the poor paint repair. You can only see swirls, scratches, dirt, micro marring, and holograms from certain angles sometimes, but would you want them on your car???