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Existence of weird complex norms

Consider the matrix $$D=\begin{pmatrix}1&0\\0&e^{i\theta}\end{pmatrix}.$$ For the commonly used norms $\|\cdot\|$ on $\mathbb{C}^2$ or for $\theta=0$ the associated subordinate norm is $1$. Is it always true ? can a subordinate norm be strictly bigger than one ?