What happens if to postulate that complex numbers whose argument differs by $2 \pi$ are not equal?

Question

What happens if to postulate that complex numbers whose argument differs by $2\pi$ are not equal? What properties such system will have? Will all analytic functions be entire?

Answer 1

Then you would have the number describing the half circle or the right angle a different name. In standard terms, the right angle is at $\frac\pi2$ and the inner angle of the equilateral triangle is $\frac\pi3$.

The definition of $\pi$ is via the right angle, in that it is the smallest positive solution to $\cos x=0$, or $e^{ix}=i$. And since the fourth power of that equality is $e^{i4x}=e^{i2\pi}=1$, you get the periodicity that you put into question.

Answer 2

Your question leads to the fascinating subject of Riemann surfaces. Riemann was motivated by the fact that some functions, such as the square root function, are not well-defined over $\mathbb C$, because each non-zero complex number has two square roots. So he introduced the notion of a Riemann surface, which in the case of the square root function is a double copy of $\mathbb C$ (except for $0$, which occurs once only).

The surface for the square root function can be visualised as a spiral staircase that joins up with itself after two revolutions. On this surface, a square root function can be defined which is single-valued and differentiable. And you have to go around the origin twice to get back to where you started, so numbers whose arguments differ by $4\pi$ (but not $2\pi$!) are equal.