Find the splitting field of $x^4+1$ over $\mathbb Q$.
This seems to be OK. Another, more general approach, is to look at this and noting that $x^4+1$ is the cyclotomic polynomial $\Phi_8(x)$, hence $x^4+1=(x-\zeta)(x-\zeta^3)(x-\zeta^5)(x-\zeta^7)$, where $\zeta \in \mathbb{C}$ is a primitive root of unity, with $\zeta^8 =1$. The splitting field over $\mathbb{Q}$ of your polynomial is $\mathbb{Q}(\zeta)$, which is indeed $\mathbb{Q}(i,\sqrt{2})$.