Is it possible to create a provably un-alterable contract?
When you send BTC to 1XXXXXXXXXXXXXXXXXXkfjds you know that nobody could possibly possess a private key to that address, so the money is considered forever lost. It has some applications, ie. Proof of Burn. However, you can't create an Ethereum contract without a private key, so any existing Ethereum contract could conceivably be altered and any associated Ether transferred away.
Let's say I wanted to make an Ethereum app for drunk gamblers called "Send me my money back in 12 hours". Basically any amount you send to it gets automatically sent back to you when you're sober (depending on how much you've drunk, YMMV). There's always the risk that the creator of the contract decides to just withdraw the Ether and screw you over.
It would be nice if you could "lock" a contract permanently, that is - creating it with a flag set that will permanently disallow any withdrawals and modifications to the contract code. Basically a public address where a private key was used to create it but any future uses of that private key would be ineffective.
Silly example I know but I'm sure there are lots of other examples where a well-designed and bug tested contract can simply exist on the blockchain autonomously without fear of the original creator changing his mind.