I get why it is interesting and useful to write complex numbers in '+' notation rather than the conventional way to denote a 2d vector, like a tuple of components.

The benefit is that multiplication and distributive property is a beauty in the '+' notation, no special rules need to be memorized for multiplying 2d vectors, i*i = -1 takes care of it.

On the other hand I never understood what the benefit, of writing the tuple of wedge and dot products in '+'notation, is.

Perhaps I am not being fair, that it is the same idea and I have not used it as much as I have used complex numbers.

More or less agreed. I think though that one reason the geometric product is so tempting is that if you take matrix representations of all of these objects, then the geometric product is literally just straightforward matrix multiplication.

Because of that, it just becomes so tempting to try and phrase everything you can in terms of this geometric product. I'm very sympathetic to the temptation, and I even think the geometric product has some great uses (it shows up a lot in some physics I do), and using it makes writing rotations a treat, but I think it's still vastly overemphasized by GA people.

I still don't really know what my favoured notation for differential geometry is, I find myself switching around so much.

Interesting. To summarize your argument: the current state of Algebra is like an 80 point solution, but to push it a few points higher requires an enormous cognitive load, and the question is whether that's really worth it, even from an educational perspective. As mentioned in another comment, this is exactly the kind of issue that comes up in Rust discussions. It seems the argument from the GA camp is that top tier mathematicians are already using these tools just fine without needing to talk about it in that way, so there's no reason for it to become general purpose. Thank you for explaining it in a way that's easy to understand. But on the other hand, maybe anomalies like these could actually become generally useful concepts. Thanks for the comment. upvoted!