[jacobolus]

A complex number is a “complex” consisting of a scalar part (“real” or “parallel” number) and a bivector part (“imaginary” or “perpendicular” number) complected together into one single object.

IMO the best way to think of it is as the quotient or ratio between two arbitrary vectors T = v/u in a Euclidean vector space. Or equivalently, as a type of mathematical operator which you can apply (by multiplication on the left) to one vector u to produce another vector v, i.e. Tu = (v/u)u = v. This operator T will scale and rotate any other vector in the plane spanned by your two vectors u and v in the same way.

http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf http://geocalc.clas.asu.edu/pdf/GrassmannsVision.pdf