The authors study the spectrum of a random perturbation of a bidiagonal Toeplitz matrix. The perturbation matrix has its entries given via independent and identically distributed complex Gaussian random variables, following the standard complex Gaussian law. The perturbation goes with a nonnegative coupling constant, which assumes very small values. The main result describes the average density of eigenvalues of the random perturbation in the interior of certain confocal ellipses.