**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.