Being $z=x+yi$ how can I factorize the polynomial $z^4+1$ as a product of real quadratic polynomials? Being $z=x+yi$ how can I factorize the polynomial $z^4+1$ as a product of real quadratic polynomials?

I don't really understand what I am asked to do. How can I start with it?

So you understood the hint
\begin{eqnarray*}
(z^2+\alpha z +1)(z^2-\alpha z +1)=z^4+\underbrace{(2-\alpha^2)}_{2-\alpha^2=0}z^2+1.
\end{eqnarray*}
To do the next one in your comment ... Factorise $z^6+1=(z^2+1)(z^4-z^2+1)$
\begin{eqnarray*}
(z^2+\alpha z +1)(z^2-\alpha z +1)=z^4+\underbrace{(2-\alpha^2)}_{2-\alpha^2=-1}z^2+1.
\end{eqnarray*}
So
\begin{eqnarray*}
z^6+1=(z^2+1)(z^2+\sqrt{3}z+1)(z^2-\sqrt{3}z+1).
\end{eqnarray*}
