The human brain must be something more than a Turing Machine.

I think Roger Penrose would agree with you on this one.

A human person is able to recognize truths that cannot possibly be logically derived by a Turing Machine. An example of this is Gödel's Incompleteness Theorem. A human can also experience love, anger, understanding, self consciousness, and the like; but it is hard to imagine how a Turing Machine could experience these.

If all of thinking takes place in the human brain, so that the mind and the brain are really the same thing, then how is this possible? It means that there must be things going on in the brain that a Turing Machine could never do. It also means that we can never build a computer that could completely simulate all the workings of the human brain.

Penrose predicts that someday we may discover how consciousness arises in the brain. However, to do so, we'll need a new theory to bridge the gap between quantum physics and classical physics.

In the preface to the 1999 printing of his book The Emperor's New Mind (Oxford University Press, 1989), Roger Penrose explains how the anesthesiologist Dr. Stuart Hameroff called his attention to the fact that most plausible place for consciousness to happen is in the microtubules in the neurons of the brain, and this may very well be where the large-scale quantum coherent actions take place.

This new theory, founded by Roger Penrose and Stuart Hameroff, is called Orch OR. There is a slide show from Daily Digest that suggests that Orch OR may lead to a scientific discovery of the immortal soul.