Complex numbers and their modulus

Question

Can we cancel the modulus on complex numbers?

For example: If we have $$|x + iy| = |n + im|$$ can we simply ignore the modulus on both sides? Or is that a false assumption?

Answer 1

Of course not. $|-1|=|1|$, and $1\neq -1$

Answer 2

No, you can't.

Consider pairs of diametrically opposite points on the unit circle, for example.

Answer 3

No, because two complex numbers can be different in coordinates, but have the same modulus.

Example: $z=1+i, w=1-i$.

Answer 4

There are an uncountably infinite number of complex numbers equal to a given valid modulus (for different values of the argument), so no.