> "Like probability theory, but over the complex numbers".

I'm a probability novice. How would this work? Are we talking vector fields representing multidimensional probabilities? Or just probabilities that happen to depend on more than one variable?

It's statistics but instead of operations preserving the 1-norm (item weights [probabilities] add up to 1) they preserve the 2-norm (squared item weights [amplitudes] add up to 1). Instead of using stochastic matrices you use unitary matrices. Everything else about quantum computing follows (...with sufficient elbow grease).

Just that if you have a quantum system, such as a qubit consisting of the |0> state and |1> state with a combination over the two (called a superposition) represented by a|0> + b|1> then a and b can be complex. This is in contrast to classical mechanics where, for example, pr(heads) and pr(tails) must both be real.

It's only once it comes to actually making a measurement that you need to get real again.

Probabilities are also always real in quantum mechanics.