Can you redownload minecraft game if you buy a different computer?

Yes. The game itself is free to download. What you paid for is the account required to play it.

So you can redownload the game for free from http://minecraft.net/...ecraft.net and log into the same account that you used before. You're in.

You'll lose the world data from the old computer and everything will be started over, but you can still play the game.