  In this paper, we study complex Jacobi matrices obtained by the Christoffel
and Geronimus transformations at a nonreal complex number, including the
properties of the corresponding sequences of orthogonal polynomials. We also
present some invariant and semi-invariant properties of Jacobi matrices under
such transformations. For instance, we show that a Nevai class is invariant
under the transformations in question, which is not true in general, and that
the ratio asymptotic still holds outside the spectrum of the corresponding
symmetric complex Jacobi matrix but the spectrum could include one extra point.
In principal, these transformations can be iterated and, for example, we
demonstrate how Geronimus transformations can lead to $R_{II}$-recurrence
relations, which in turn are related to orthogonal rational functions and
pencils of Jacobi matrices.
