Civilization 4 was a little different with where you might want to settle, and in Civilization 5, there is no real benefit of even settling next to a river for your capital. Sure, you will lose out on some water buildings, but overall, they aren't worth the hammers they cost to build them anyway.
So like this question which asked how important it is to settle next to a river, I'm curious: If your Settler spawns on a hill, is there ever any reason to move it to another tile?
Correct me if I am wrong, because I have only completed one settler level game and now I'm running immortals and deity, but doing the math, a hill tile is always better. You get the additional hammer for production early on, which means you can strike out a Monument 3 turns earlier. The earlier Monument means that if you go the Pottery > Writing route, you can slingshot into Medieval by getting the Great Library.
I restart all of my games at about turn 100, but from what I can gather, settling on a hill means I get Liberty earlier, and get my Settler about 7 turns earlier.
Is a hill start the best start you can get?
Is it worth wasting a turn to settle on a hill?