At the very first day of my complex analysis class, I learned that for any complex number $z=x+iy$ there exist another complex number given by $\overline{z}=x-iy$ such that both $z+\overline{z}=2x$ and $z\overline{z}=x^2+y^2$ are real.
I curious that, whether can we extend this notion for more than two complex numbers?
More explicitly, for a given $z\in\mathbb{C},$ can we find non trivial $z_1,z_2\in\mathbb{C}$ such that all $$z+z_1+z_2,$$ $$zz_1+zz_2+z_1z_2$$ and $$zz_1z_2$$ are real? and so on.
Hint: three complex numbers are roots of a 3rd degree polynomial with real coefficients only if one them is real, and the other two are either real or complex conjugates.