No the fee is not fixed but the amount has no bearing on the fee. It doesn't matter if you send .0001 btc or 10000 bitcoin the fee could be the same. I say could be because the fee is based on the size of the transaction in bytes. The more inputs and outputs it has the more it costs. Most transactions will only be 1 or 2 inputs and 2 outputs though. Still even really large transactions aren't that expensive but would be a lot less expensive without the current artificial blocksize limit. A year ago just about any transaction could be done for under 3 cents.