Characterizing complex solutions of $\frac{1}{z_1}+\frac{1}{z_2}=\frac{1}{z_1+z_2}$ What I have done so far isn't so helpful. If $z_1=x_1+y_1i$ and $z_2=x_2+y_2i$ then these two things must hold:
$$x_1^2 -y_1^2 +x_2^2-y_2^2+x_1x_2-y_1y_2=0$$
$$2x_1y_1+2x_2y_2+x_1y_2+x_2y_1=0$$
This also means that
$$\Re(z_1z_2)=y_1^2+y_2^2-(x_1^2+x_2^2)$$
$$\Im(z_1z_2)=-2(x_2y_2+x_1y_1)$$
But again, this isn't very helpful. Are there any simpler solutions that don't involve this nasty algebra?

As long as $z_1,z_2,z_1+z_2\neq 0$, you can cross-multiply to get an equivalent equation $z_2(z_1+z_2)+z_1(z_1+z_2)=z_1z_2$, which rearranges to $$z_1^2+z_1z_2+z_2^2=0.$$
By the quadratic formula you can solve for $z_1$ in terms of $z_2$: $$z_1=\frac{-z_2\pm\sqrt{z_2^2-4z_2^2}}{2}=\frac{-1\pm i\sqrt{3}}{2}z_2.$$
So your original equation has all the solutions to this except for those for which $z_1$, $z_2$, or $z_1+z_2$ are $0$ (in fact, you can check that the only way any of them can be $0$ is if $z_1=z_2=0$).
