Absolute value of complex numbers

[variety]

When is it true that is |a|=|b|, then either a=b or a=b*, where a and b are complex numbers and b* is the complex conjugate of b?

 

[CompuChip]

-1 is not the complex conjugate of 1.

 

[variety]

What I'm asking is, are there any conditions on a and b such that the above statement is true?

 

[CompuChip]

Yes, but they are all trivial rewritings of |a| = |b|.
If it were as simple as "a = b or a = b*", for example, we wouldn't need the bar notation, would we

Some of those rewritings / interpretations are:
* a / b lies on the unit circle
* a and b lie on a circle in the complex plane
* the distance of a and b to the origin is the same
* Re(a)^2 - Re(b)^2 = Im(b)^2 - Im(a)^2
* There exists r > 0, \theta_1, \theta_2 \in [0, 2\pi) such that a = r e^{i\theta 1}, b = r e^{i\theta_2} (and r = |a| = |b|)
* There exists \theta \in [0, 2\pi) such that a = e^{i \theta} b.