  It is known that the generating function $f$ of a sequence of Toeplitz
matrices $\{T_n(f)\}_n$ may not describe the asymptotic distribution of the
eigenvalues of $T_n(f)$ if $f$ is not real. In a recent paper, we assume as a
working hypothesis that, if the eigenvalues of $T_n(f)$ are real for all $n$,
then they admit an asymptotic expansion where the first function $g$ appearing
in this expansion is real and describes the asymptotic distribution of the
eigenvalues of $T_n(f)$. In this paper we extend this idea to Toeplitz matrices
with complex eigenvalues. The paper is predominantly a numerical exploration of
different typical cases, and presents several avenues of possible future
research.
