Minimal polynomial of $\sqrt3 + i$ over $\mathbb{R}$ and $\mathbb{C}$ I was able to show that $x^4-4x^2+16$ is the minimal polynomial of $\sqrt3 + i$ over $\mathbb{Q}$.
Does this imply that this is also the mininal polynomial over $\mathbb{R}$ and $\mathbb{C}$?

No. Every polynomyal in $\mathbb R[X]$ can be factored in linear factors and quadratic factors depending on how many roots are real and how many are not, thus the minimum polynomial has degree at most $2$. In this case it is of degree 2, ie $(x-\alpha)(x-\overline{\alpha})$. Note that when there is a real solution $\beta$ you can factor out $(x-\beta)$ and repeat, when there is a not real root $\gamma$ ie $p(\gamma)=0$ you can see that $p(\overline{\gamma})=\overline{p(\gamma)}=0$ also, thus you can factor out $(x-\gamma)(x-\overline{\gamma})$ which is of course in $\mathbb R[X]$.
Every polynomial in $\mathbb C[X]$ can be factored in linear factors because every polinomyal of degree at least $1$ has a solution via the Fundamental Algebra Theorem. Thus the minimum polynomial has degree $1$, ie $(x- \alpha) \in \mathbb C[X]$.
