Yeah rotation matrix are not hermitian (note that you don't need to show the eigenvalues aren't real, you just need to show it's not self-conjugate). The OP may

Yeah rotation matrix are not hermitian (note that you don't need to show the eigenvalues aren't real, you just need to show it's not self-conjugate). The OP may have been confused by the fact that you can make a rotation matrix of eigenvectors of a hermitian matrix, which diagonalises the original matrix into two conjugate rotation matrices with a diagonal matrix between them.