Every complex normal matrix has a choice of orthonormal eigenvector basis. By the spectral theorem $A=UDU^*$; the columns of $U$ supply such a basis. Arbitrary eigenvectors need not be normalized or mutually orthogonal within a repeated eigenspace.
Example: $A=\operatorname{diag}(1,i)$ has the orthonormal eigenvectors $e_1,e_2$. Distinct eigenspaces are orthogonal. Within one eigenspace, apply Gram–Schmidt to select an orthonormal basis.
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