Determining locus of a complex number

The given statement implies that $z$ is equidistant from $-\bar{z}$ and $\bar{z}$ so it lies on the perpendicular bisector of $\ldots$

But $\,z\,$ is the variable in this case, so the perpendicular bisector of $\,\bar z\,$ and $\,-\bar z\,$ changes for each $\,z\,$.

Instead, use that $\,z+ \bar z = 2 \operatorname{Re}(z)\,$ and $\,z- \bar z = 2i \operatorname{Im}(z)\,$, then the equality reduces to:

$$\require{cancel} |\operatorname{Re}(z)|=|\operatorname{Im}(z)| $$

The latter is equivalent to $\,\operatorname{Re}(z)=\pm \operatorname{Im}(z)\,$, or $\,x=\pm y\,$ in Cartesian coordinates.

$z$ is not a constant in this problem... thus neither are $\bar z$ and $-\bar z$...
That's where your error is.

For $z$ to satisfy the equation $\lvert z+\bar z\rvert=\lvert z-\bar z\rvert$, we need only $x=\pm y$.

Note that

$$|z+\bar z|=|2Re(z)|$$

$$|z-\bar z|=|2Im(z)|$$

thus the locus $$|2x|=|2y|\iff|x|=|y|$$

is correct.

Indeed we can also write

$\left|\frac{z+\bar z}2\right|=\left|z-\frac{z-\bar z}2\right| \equiv$ distance of z from y axis

$\left|\frac{z-\bar z}2\right|=\left|z-\frac{z+\bar z}2\right|\equiv$ distance of z from x axis

thus the locus we are looking for are the bisectors.