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.