Is there a known name for the commutation property in the complex absolute value formula? The supposed property can be resumed as follows:
The absolute value of a complex number remain unaltered when real and imaginary parts of complex number commute, $e.i$:
$|a + ib| = |b + ia| = a^2 + b^2$
may this be related to the fact that the absolute value for the imaginary unit $i$ and its mapping $i^*$ need to have the same absolute value?
$|i| = |i^*| = 1$?
The mapping $a+bi \to b+ia$ is actually just a reflection over the line $\{x+xi| x\in\mathbb R\}$, and because this line goes through the origin (through $0$), the distance from any point to zero remains the same after performing the reflection.
For any two points, $z_1$ and $z_2$, no reflection can change the distance between them, meaning that if $z_1$ maps to $z_1^*$ and $z_2$ maps to $z_2^*$, then the distance between $z_1$ and $z_2$ is the same as the distance between $z_1^*$ and $z_2^*$.
This means that in your case, because $0$ maps to $0^*$, you know that for any point $z$, the norm of $z$ (which is the distance between $z$ and $0$) must be the same as the distance between $z^*$ and $0^*$ (which is the norm of $z^*$).