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 $1/2$ and the complex norm of $A$ is $1$.