[DoctorOetker]

For me personally, I prefer to perceive as more fundamental those constructions that straightforwardly generate the most with the least of novelty.

For example one may be introduced to the real numbers, and later to the complex numbers and then later perhaps the quaternions and the octonions, etc. in a haphazard disconnected way.

Given just the real numbers and "geometric algebra" (i.e. Grassmann algebras), this generates basically all these structures for different "settings".

I hence view geometric algebra as a lower level and more fundamental than complex numbers specifically.

A more interesting question (if we had to limit discussion to complex numbers) would be the following: which representations of complex numbers are known, what are their advantages and disadvantages, and can novel representation of complex numbers be devised that display less of the disadvantages?

For example, we have -just to list a few- the following representations:

1. cartesian representation of complex numbers: a + bi : the complex numbers permit a straightforward single-valued addition and single-valued multiplication, but only multivalued integer-powered-roots. addition and multiplication change smoothly with smooth changes in the input values, taking N-th roots do not, unless you use multivalued roots but then you don't have single valued roots.

2. polar representation of complex numbers: r(cos(theta)+i sin(theta)): this representatin admits smooth and single valued multiplication and N-th roots, but no longer permits smooth and single valued additions!

Please follow up with your favourite representations for complex numbers, and expound the pro's and con's of that representation.

For example can you generate a representation where addition and N-th roots are smooth and single valued, but multiplications are not?

Can you prove that any representation for complex numbers must suffer this dilemma in representation, or can you devise a representation where all 3 addition, multiplication, roots are smooth and single valued?