Suppose $D={\rm diag}(d_i)$ is a diagonal matrix with all diagonal entries $d_i=\pm 1$. This implies $D^2=I$. Suppose $A$ is a non-singular Hermitian matrix. If we know that $A+A^{-1}+D$ has rational eigenvalues, what can we say about eigenvalues of $A$?

If the question suggests that then $A$ also should have rational eigenvalues, then it is easy to produce counterexamples. For, say, $$ A=\begin{pmatrix} 1 & 1 \cr 1 & 2 \end{pmatrix}, \; D=\begin{pmatrix} 1 & 0 \cr 0 & 1 \end{pmatrix} $$ we obtain $A+A^{-1}+D=3D+D=4D$, which has integer eigenvalues. The eigenvalues of $A$ are $\frac{3\pm \sqrt{5}}{2}$.