Fix some base field $F$ and some $\alpha,\beta$ such that $F(\alpha) = F(\beta)$. We know that there exists some minimal polynomial $p(x)$ such that $\alpha$ satisfies $p(x)$ and any polynomial in $F[x]$ which $\alpha$ satisfies is divisible by $p(x)$ (and the same for $\beta$ and some $q(x) \in F[x]$). My question is what can we say about the relationship between two elements (or perhaps their minimal polynomials) which generate the same extension over the same field. Obviously they have the same degree, but is there anything more we can say?
One case I was thinking about/discussing with some friends is $\mathbb{Q}(i)$ and $\mathbb{Q}(i+1)$. These clearly yield the same extension of $\mathbb{Q}$, but $i$ has minimal polynomial $x^2+1$ and $i+1$ has minimal polynomial $x^2-2x+2$, but I can't see a relationship between the two.
Note I'm not interested in the special case where $\alpha$ and $\beta$ are distinct roots of the same minimal polynomial, as I understand that situation okay already.