Confusion about roots Solving $$z^4 + 1 = 0$$ where $z = x+iy$, I tried by factorizing the expressiong into $$z^4 + 1 = (z^2 - 2z + 2)(z^2 + 2z + 2).$$ But then solving $$z^2 - 2z + 2 = 0$$ for $z$ gives the solutions $z = 1 \pm i$, and solving $$z^2 + 2z + 2 = 0$$ for $z$ gives the solutions $z = -1 \pm i$. The actual roots should be $z = \frac{1 \pm i}{\sqrt{2}} $ and $ z= \frac{-1 \pm i}{\sqrt{2}}$. I'm possibly missing something that should be obvious to me here, but what about $1/\sqrt{2}$?

Your starting factorization is not correct. It should be: 
$$z^4+1 = z^4+2z^2+1-2z^2 = (z^2+1)^2-(\sqrt{2}z)^2=...$$
