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 $1$.
Iosif Pinelis
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