Abstract: We present recent numerical observations concerning localization of eigenvalues of sequences of certain complex Jacobi matrices. These matrices are (essentially) non-hermitian which makes the analysis particularly difficult (standard results of perturbation theory for self-adjoint operators are not applicable, estimates on the norm of resolvent are not available, etc.). As a consequence, many interesting mathematical questions arise. We provide several of them as conjectures and open problems. We give a comment on a possible approach solving these problems, show that a similar phenomena appears in case of Toeplitz matrices and indicate some inclusions in the potential theory.