Question

We know for a fact that the equation:

Answer 1

They don't need to be pure real or imaginary. Any "mixed" complex number works so long as $z=z'$.

Let $z=z'=a+bi$. Then

$|z+z'| = |2a+2bi| = 2 \sqrt{a^2+b^2}$

$|z| + |z'| = 2|a+bi| = 2 \sqrt{a^2+b^2}$

so $|z+z'|=|z|+|z'|$.

The geometric interpretation is essentially identical. $|z+z'|=2|z|$ is a complex number twice the distance away from the origin in the complex plane as $z$, which is exactly $|z|+|z|$.