$x^{4}+1$ needs to be factored into the product of two trinomials, and based on this, the equation $x^{4}+1=0$ needs to be solved.
The given expression can also be written as:

$$
x^{4}+2 x^{2}+1-2 x^{2}=\left(x^{2}+1\right)-2 x^{2}=\left(x^{2}+x \sqrt{2}+1\right)\left(x^{2}-x \sqrt{2}+1\right)
$$

Thus, the roots of the equation $x^{4}+1=0$ are

$$
x_{1}=\frac{-\sqrt{2} \pm \sqrt{-2}}{2}, x_{2}=\frac{\sqrt{2} \pm \sqrt{-2}}{2}
$$

(Zoltán Harsányi, Eger.)

The problem was also solved by: Déri Zs., Haar A., Kertész G., Kürti I., Messer P., Söpkéz Gy.