I was told today by a friend that having a zero along there main diagonal of a matrix will promote complex eigenvalues. I do not believe this is true because the below matrix Z has a zero present along the diagonal and yet has all real eigenvalues.
$Z=\begin{bmatrix} 0 & 0 & 0 & 0\\ 0 & 2 & 0 & -1\\ 1 & 0 & -1 & 0\\ 2 & 0 & 0 & -1\\ \end{bmatrix}$
Besides proving this statement incorrect with an empirical example, is there any significance to zeros being along the diagonal of a square matrix with respect to eigenvalues?
Your counter example is correct (you could also consider the zero matrix, it has lots of zeros everywhere, but no non-real eigenvalues). Remember that similar matrices have the same eigenvalues. However, zeros here or there in a matrix does not say much about zeros here or there in all matrices similar to it. Thus, you can't expect much from seeing a zero on the main diagonal as far as qualitative information about the linear transformation.