[dragontamer]

> By the way, an interesting question is how complex boolean numbers could be defined (naturally).

Ever work with GF(2^x) extension fields?

For a good practical application of "binary complex numbers" (aka: GF(2^x) extension fields), see Reed Solomon error correction codes. EDIT: NASA's Reed Solomon code tutorial is an excellent, "casual", introduction to the subject. (Very little math theory involved: mostly sticks to the "final math" so to speak, focusing on the Reed Solomon error correction aspects). Once you understand how Reed Solomon codes are used, you can then more easily go into the theory of how it relates to GF extension fields, and how those GF-extension fields are similar to complex numbers made up of 0s and 1s.