Writing the equation as $|z - |z + 1||^2 = |z + |z - 1||^2$, and using $|w|^2 = w \cdot \bar{w}$, for any complex number $w$, yields the equivalent form
$$(z + \bar{z}) (|z - 1| + |z + 1| - 2) = 0.$$
We deduce that either $z + \bar{z} = 2 \operatorname{Re} z = 0$, hence $z = ia$, for some real $a$, or $|z - 1| + |z + 1| = 2$. In this second case, we deduce that, in the complex plane, the sum of distances from the point $z$ to the points $-1$ and $1$ equals $2$, which is possible if and only if $z$ is a real number, lying on the line segment $[-1, 1]$.