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Over $\mathbb{C}$, are unitarily and orthogonally diagonalizable the same?

I know that unitary matrices are not necessarily orthogonal, but I can't find a counterexample to unitarily and orthogonally diagonalizable being the same. (Unitarily diagonalizable: $A=U^{-1}DU$ where $U$ is unitary and $D$ is complex and diagonal; replace unitary with orthogonal for orthogonally diagonalizable)

(Orthogonal: real unitary matrix)

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    $\begingroup$ To be clear, what do you mean by "orthogonal" here? A unitary that has real entries? Or a complex matrix $O$ such that $O^T=O^{-1}$? $\endgroup$
    – Eric Wofsey
    Commented Dec 14, 2021 at 3:32

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Consider an orthogonal matrix (in $SO(2),$ if you like). Any such matrix is unitarily diagonalizable, and almost no such matrix is orthogonally diagonalizable (if it were, its eigenvalues would be $\pm 1.$

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  • $\begingroup$ This seems to only apply when $D$ is real. $\endgroup$
    – my2cents
    Commented Dec 14, 2021 at 3:29

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