Given that the minimal polynomial for both elements is $X^4 -2$, I imagine that the fields are isomorphic but I'm finding it difficult to find a bijection between the fields (if that is indeed the approach I need to take).
I know an isomorphism $\varphi : \mathbb{Q}(\sqrt[4]{2}) \rightarrow \mathbb{Q}(i\sqrt[4]{2})$ needs to fix $\mathbb{Q}$ so I was considering something along the lines of complex conjugation, but I've hit a dead end with that approach.