An ellipse via a modulus equation

Interpretation. $|z-1|$ is the distance from $M$ to $A(1,0)$ and $|z+1|$ is the distance from $M$ to $B(-1,0)$. The equation states that $MA+MB=4$.

Ellipse. The set of points whose sum of distances to two fixed points is constant is an ellipse with foci $A$ and $B$. The constant is $2a=4$, so $a=2$; the focal distance is $2c=2$, so $c=1$. Then $b^2=a^2-c^2=3$.

Conclusion. The locus is the ellipse with foci $(\pm1,0)$, semi-major axis $2$, semi-minor axis $\sqrt3$: $\frac{x^2}{4}+\frac{y^2}{3}=1$.