Too Complex??

Note that the summation can be manipulated and written as

$z_1^3+z_2^3+z_3^3-3z_1z_2z_3=-4z_1z_2z_3$

Now, factorising the left side, we get

$\left(z_1+z_2+z_3\right)\left(\left(z_1+z_2+z_3\right)^2-3\left(\sum_{z_1\ ,\ z_2\ ,\ z_3}z_1z_2\right)\right)=-4z_1z_2z_3$

Now, because $\displaystyle z\cdot\left(\sim z\right)=\left|z\right|^2$ for any complex number $\displaystyle z$ we have

$\left(z_1+z_2+z_3\right)\left(\left(z_1+z_2+z_3\right)^2-3z_1z_2z_3\left(\sim\left(z_1+z_2+z_3\right)\right)\right)=-4z_1z_2z_3\$

Now, putting $\displaystyle z=z_1+z_2+z_3$ we get

$z^3=\left(3\left|z\right|^2-4\right)z_1z_2z_3$

Finally, taking modulus on both sides gives us

$\left|z\right|^3=\left|3\left|z\right|^2-4\right|$

Solving this equation gives us $\displaystyle \left|z\right|=-2,1,2$ . But since $\displaystyle \left|z\right|\ge0$ , we have

$\left|z_1+z_2+z_3\right|=1,2$

Note:

Here $\displaystyle \sim z$ denotes the conjugate of the complex number $\displaystyle z$