$\newcommand\R{\mathbb R}\newcommand\C{\mathbb C}$No. E.g., let $$A=\begin{bmatrix}1&0\\0&0 \end{bmatrix}$$ and $$\left\|\begin{bmatrix}z_1\\z_2 \end{bmatrix}\right\|=|z_1|+|z_1+iz_2|.$$ Then the real norm of $A$ is $1$ and the complex norm of $A$ is $2$.
Indeed, if $x=\begin{bmatrix}x_1\\x_2\end{bmatrix}\in\R^2$ with $x_1\ne0$, then $$\frac{\|Ax\|}{\|x\|}=\frac{|x_1|+|x_1+0i|}{|x_1|+|x_1+ix_2|} \le\frac{2|x_1|}{|x_1|+|x_1|}=1,$$ with $\frac{\|Ax\|}{\|x\|}=1$ if $x_2=0$. So, the real norm of $A$ is $1$.
If now $z=\begin{bmatrix}z_1\\z_2\end{bmatrix}\in\C^2$ with $z_1\ne0$, then $$\frac{\|Az\|}{\|z\|}=\frac{2|z_1|}{|z_1|+|z_1+iz_2|} \le2,$$ with $\frac{\|Az\|}{\|z\|}=2$ if $z_2=iz_1$. So, the complex norm of $A$ is $2$.
