I'm not sure if this property is standard but this is what some examples suggested:
Let $\alpha$ and $\beta$ be algebraic numbers. Is it true that $$|N_{\mathbb Q(\alpha) / \mathbb Q}(min_{\beta / \mathbb Q}(\alpha))| = |N_{\mathbb Q(\beta) / \mathbb Q}(min_{\alpha / \mathbb Q}(\beta))| ?$$
I would be really obliged if someone could provide a proof or counterexample. If the above conjecture is false, can something be said in the case in the following special cases:
When the extension $\mathbb Q(\beta) / \mathbb Q$ is Galois?
When the extension $\mathbb Q(\beta) / \mathbb Q$ is a cyclotomic extension?
Let $S,T$ be the (finite) sets of embeddings of $\mathbb{Q}(\beta)$ and $\mathbb{Q}(\alpha)$ in $\mathbb{C}$. Let $n=|S|$, and, for each $0 \leq k \leq n$, $(-1)^kb_k$ be the $k$-th elementary symmetric polynomial in the elements of $S$.
Then $min_{\beta/\mathbb{Q}}(\alpha)=\prod_{\sigma \in S}{(\alpha-\sigma(\beta))}=\sum_{k=0}^n{\alpha^kb_{n-k}}$.
So the norm $\mathbb{Q}(\alpha)/\mathbb{Q}$ of the LHS is $\prod_{\tau \in T}{\sum_{k=0}^n{\tau(\alpha)^kb{n-k}}}=\prod_{\tau \in T}{\prod_{\sigma \in S}{(\tau(\alpha)-\sigma(\beta))}$.
So actually, the quotient $LHS/RHS$ for you is equal to $(-1)^{|S||T|}$ and the equality holds iff $\alpha$ and $\beta$ are conjugates or $|S|$ or $|T|$ is even. Otherwise, there is a sign.