For each u in C independent from 1 (so that C is spanned by 1 and u), the mapping x+yu → -y+xu is an orthonormality which maps 1 to u.

After the introduction of such an operator ω, and having defined i:=ω(1), and the conjugate of x+yi as conj(x+yi):=x-yi,
a roto-homothety is quickly defined as a map ρ:C→C such that:
I) ρ(i) is orthonormal to ρ(1), i.e. ρ(i)=ω(ρ(1)), that ρ(ω(1))=ω(ρ(1))
II) ρ(x+yi)=ρ(x1+yi)=xρ(1)+yρ(i), i.e. the coordinates of a point remain the same (this is the etymological meaning of "homothety")
and ρ is defined to be a rotation if it maps conj(ρ(1)) to 1 (at the same time that it maps 1 to ρ(1)).
So all the goniometrical theory can be derived from here.

Aha, I see. You develop an axiomatic theory for which the common complex numbers serve a model. You give a fine definition of rotation.

Your claim is that this is a more natural way of looking at complex numbers. Is that right?

I undertsand so far.

>That makes me think, once again, about the
>Goedel's incompleteness theorem.

Why?

>An axiomatic framework for developing plane analytical
>geometry can be carried out by means of an "octet" structure
>(C , + , 0 , - , R+, · , 1 , i )

This I do not understand. I believe your axiomatics made use of the whole R; C was always a pair of real numbers.

>Why?

The fact that we cannot "catch" the "optical" orthogonality by the geometrical Hilbert set of axioms (neither, of course, from its vectorial counterpart) reminds me the Goedel's incompleteness theorem.

* * *
>>An axiomatic framework for developing plane analytical
>>geometry can be carried out by means of an "octet" structure
>>(C , + , 0 , - , R+, · , 1 , i )

>This I do not understand. I believe your axiomatics made use of the whole R; C was always a pair of real numbers.

in
https://w3.romascuola.net/gspes/c/lo/1.html
the set R+ is defined to be as a subset of C (which is not defined by means of R, but as a set to be "axiomatically defined") with the properties given in
https://w3.romascuola.net/gspes/c/lo/3b.html .
Of course these are "almost" the usual "real line" topological and order properties, but stated in a superset (i.e. C) environment.
A remarkable (and well known from technical design) fact is that the drawing of the sum and product of two real numbers cannot be carried out without the help of a point outside from the unidimensional world of R itself, just because parallelograms (for the sum) and proportional triangles (for the product) need two dimensions. So C is a more suitable environment to graphically work on real numbers.
The minimal topological request to built C is to have a subset of C which allows a linear multiplication (i.e. a set of scalar coefficients for product) and which translates the intuitive "zero-one ray" that, afterwards, can be used to build all the real axis (by means of the opposition symmetry), then to "build" (together with the other unit "i") the general product with complex coefficients, and (as a polar ray) to measure angles by means of the other unit "i" and the Euler "archification" of a real number k, i.e. the set {exp(k*i)=lim(1+t*i/n)^n : t varies in <0,k>}:
move c horizontally in (copy and paste this adress on a browser):
w3.romascuola.net/gspes/pug.html?z=80&c=1-i&k=cx&m=exp(k(i))&f=exp((t)k(i))

Example: (copy and paste the following parametrical url)

w3.romascuola.net/gspes/pug.htm?z=80&q=1&c=1-i/4&k=cx&n=10000&m=(1+k*i/n)^n&f=(1+t*k*i/n)^n{_move_c_horizontally_}&g=1+k*i&h=1+t*k*i

For a short explanation of the url-parameters see:

https://w3.romascuola.net/gspes/pug.html?q=1&f={write_math_here}

For Example, my proof of the PT becomes on PUG:

w3.romascuola.net/gspes/pug.htm?x=550&y=570&o=0.6+0.4i&a=1+2i&z=90&t=1000&f=((|a|+ax)(1-i)/2+(t-(floor(200t)=200t)i)(|a|-ax)/2)(ax>0)(ay>0)&g=(t-(floor(200t)=200t)i)ax(ax>0)(ay>0)&h=((|a|+ax)/2+t(|a|-ax)/2-ax(floor(200t)=200t)i)(ax>0)(ay>0)&k=(ax*t+ay*t(floor(200t)=200t)i)(ax>0)(ay>0)&m=(ax+ay/2+ay/2i+(t+(floor(200t)=200t)i)ay/2)(ax>0)(ay>0)&n=((floor(31t)=1)a+(floor(31t)=2)a(1+i)+(floor(31t)=3)a*i-i(floor(31t)=5)|a|+(floor(31t)=6)|a|(1-i)+(floor(31t)=7)|a|+(floor(31t)=8)a+(floor(31t)=9)(ax-|a|i)-(floor(31t)=10)|a|i-(floor(31t)=11)(|a|+ax)i/2+(floor(31t)=12)(|a|-(|a|+ax)i/2)+(floor(31t)=13)(|a|-ax*i)-i(floor(31t)=14)ax+(floor(31t)=16)|a|+(floor(31t)=18)((|a|+ax)/2-|a|i)+((floor(31t)=17)+(floor(31t)=19)+(floor(31t)=21))(a+|a|)/2+(floor(31t)=22)(ax+ay/2i)+(floor(31t)=23)a+(floor(31t)=24)(a+ay)+(floor(31t)=25)(ax+ay)+(floor(31t)=26)ax+(floor(31t)=27)(ax+ay/2i)+(floor(31t)=28)(ax+ay+ay/2i)+(floor(31t)=29)(ax+ay)+(floor(31t)=30)(ax+ay/2)+(floor(31t)=31)(a+ay/2))(ax>0)(ay>0)

(copy and paste in the browser, and then move the point "a")