Let $A\in M_n(\mathbb{R})$ be a matrix and $\|\cdot\|$ be a norm on $\mathbb{C}^n$. When we look at the operator norm of $A$ with respect to $\|\cdot\|$ we can either consider the inclusion of $M_n(\mathbb{R})$ in $M_n(\mathbb{C})$ or the restriction of $\|\cdot\|$ to $\mathbb{R}^n$. Are these points of view equivalent?

In other words do we always have: $$ \sup_{x\in\mathbb{R}^n\setminus\lbrace0\rbrace}\frac{\|Ax\|}{\|x\|}=\sup_{x\in\mathbb{C}^n\setminus\lbrace0\rbrace}\frac{\|Ax\|}{\|x\|} $$ for $A\in M_n(\mathbb{R})$.

I hoped that we could use as an intermediary step this other question however, I was a bit optimistic.