= Solution
Use
$$
\boxed{D=\begin{pmatrix}0&-1\\1&0\end{pmatrix}.}
$$
It has independent complex <eigenvectors> $(1,-i)^T$ and $(1,i)^T$, with <eigenvalues> $i$ and $-i$. Taking these as columns gives an invertible complex matrix $P$ with $P^{-1}DP=\operatorname{diag}(i,-i)$, so $D$ is a <diagonalizable matrix> over $\mathbb C$. It cannot be diagonalized over $\mathbb R$, since a real diagonal matrix would have real <eigenvalues>, whereas similarity preserves the <characteristic polynomial>.
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