This gets directly to my main question each time I see geometric algebra show up: how does it fit in with the "normal" notation of differential geometry? What assumptions does it make for a metric, what is the equivalent of parallel transport, Lie brackets, how does it represent gradients (and other things that are naturally 1-forms), etc., etc.?

All the treatments I've seen jump in to manipulation without really going in to the axioms used. That paper seems to do a lot better at fitting it together, so I'll certainly read it, thanks.

Thanks for sharing. At what point does something like this become a PhD thesis level topic and not "just" a paper quality topic?

I will read this a bit later, but is the idea to extended the structure of the tangent space and make it a full geometric algebra?