Suppose that $A$ is a complex matrix satisfying $A^TA = I$ (so $A$ is the entrywise transpose, not the conjugate transpose). What can be said about the eigenvalues of $A$, if $A$ is "complex-orthogonal" in this sense?
Of course, for any eigenpair $(\lambda,x)$, we have $$ x^Tx = x^TA^TAx = (Ax)^TAx = \lambda^2 (x^Tx) $$ which allows us to conclude that $\lambda^2 = 1$... so long as $x^Tx \neq 0$. Can anything else be said? Does the case in which $A$ has real entries allow us to conclude that $|\lambda| = 1$?