Let $K$ be a finite extension of $\mathbb{Q}$. For each $\alpha \in K$, multiplication by $\alpha$ is a linear map from $K$ to $K$. Fix an integral basis $\omega_1,\ldots,\omega_n$ for $K$ over $\mathbb{Q}$. Let $M_{\alpha}$ be matrix for the "multiplication by $\alpha$" map with respect to the given basis.
If $\alpha = \sum_{i=1}^{n} a_i \omega_i$, set $$\| \alpha \|_{1} = \max_{1 \leq i \leq n} \| a_i \|$$ and set $$ \| \alpha \|_{2} = \max_{1 \leq i,j \leq n} \| (M_{\alpha})_{ij} \| $$ where $(M_{\alpha})_{ij}$ is the entry in the $i$-th row and $j$-th column of $M_{\alpha}$.
Both $\| \cdot \|_1$ and $\| \cdot \|_2$ are norms on $K$, which is finite-dimensional, thus the norms are equivalent in the sense that there are constants $c,C > 0$ such that $$ c \|\alpha \|_1 \leq \| \alpha \|_2 \leq C \|\alpha \|_1 $$ for all $\alpha \in K$.
Update Since $K$ is over $\mathbb{Q}$ (as opposed to $\mathbb{R}$ or $\mathbb{C}$), it is not necessarily true that all norms on $K$ must be equivalent. So that gives a preliminary question: Are the two norms equivalent?
Update 2: I posted an answer that resolves the question in the first update. In summary, the norms are equivalent because they are equivalent when extended to the vector space given by the formal span $\text{span}_{\mathbb{R}}(\omega_1,\ldots,\omega_n)$. The questions below still stand.
Question How do the constants depend on $K$? Do they depend only on the degree $n$ of $K$, if so how? For a given degree $n$, can we always find a $K$ so that $c$ and $C$ are as small and large, respectively, as we desire?
Side Question Is there any good reference for properties of matrix representations of algebraic numbers? Every book I've seen hardly goes beyond saying that the determinant of $M_{\alpha}$ is the norm of $\alpha$.