The Cambridge University GA Research Group’s website along with the “Geometric Calculus R & D Home Page” should serve as a good introductions to geometric algebra, along with the Wikipedia reference – “Geometric algebra”. A somewhat more advanced exploration in the context of differential geometry can be found in the book titled “A New Approach to Differential Geometry using Clifford's Geometric Algebra”. A development with more of a 'universalist' flavour (universal as in universal algebra) is presented at the geocalc.clas.asu.edu page “Universal Geometric Calculus ”.
Should geometric algebra be presented to undergraduates in engineering, physics and mathematics early in their curriculum instead of, or alongside, linear algebra?
Good answers should try to analyze:
which subject is more fundamental, in an attempt to identify where they should fit in in a curriculum
what changes would be required in the general curriculum in order to meaningfully support an early introduction of geometric algebra, and whether such changes would be feasible given the current infrastructure
why such curriculum changes have not been made thus far
examples of personal or institutional experiences resulting from an attempt to make such a curriculum change (if any exist)
whether a problematic or beneficial slippery slope exists -- could an (escalated, seemingly more "extreme") argument be made for the introduction of "applied abstract algebra" early in high school as a consequence of serious consideration given to the main question?