Consciousness - as far as Harris seemed to be using it - is just subjective awareness of qualia. It could be immaterial yet attached to matter (I believe this is formally known as property dualism), which I think it what Harris was hinting at.
Or it could be running the right program (I doubt this), or some kind of Awareness into which all identity dissolves upon death of the body, etc. So there could be consciousness of an immaterial variety without necessarily changing the relationship between mind & brain.