Equation in the complex field $(z+2)^6=z^6$ I'm wondering why if $z$ is a solution of the equation
$$
         (z+2)^6=z^6
$$
then we must have $\Re(z)=-1$. I've tried to take the real part of both sides, noticing that
$$
 \Re((z+2)^6)=\Re([(z+2)^3]^2)=|(z+2)^3|^2
$$
but it doesn't seem to work. Thank you in advance.

Doing $z = w - 1$:
$$0 = (z + 2)^6 - z^6 = 4w(3w^4 + 10w^2 + 3),$$
$w = 0$ ($z = -1$) is obviously solution and the biquadratic factor $3w^4 + 10w^2 + 3$ has only purely imaginary solutions because $w^2 = -3,-1/3$.
