It seems very interesting, but I have not currently the time to make a detailed comparison with vector/tensor algebra. Moreover I have not

I would suggest the freely available Oersted Medal Lecture 2002 by David Hestenes http://geocalc.clas.asu.edu/pdf/OerstedMedalLecture.pdf the person who discovered and developed the Geometric Algebra. In particular, see Section V of the above paper. It talks about vectors, geometric products, the coordinate-free representation for `vector product' and the geometric meaning of the imaginary unit i. Section 1 gives a good motivation for the Geometric Algebra. Other sections of the paper develop physical applications, from classical mechanics to electrodynamics to non-relativistic and relativistic quantum mechanics. Computer Scientists might then like http://www.geometricalgebra.net/ see the good and free introduction http://www.geometricalgebra.net/downloads/ga4cs_chapter1.pdf Incidentally, David Hestenes said in the lecture that he has applied for an NSF grant to work on Geometric Algebra TWELVE times in a row, and was rejected every single time.