  This paper is a supplement to and extension of arXiv:1903.01399. In the
internal twistor space of the 4D Vasiliev's higher-spin gravity, we study the
star-product eigenfunctions of number operators with generic complex
eigenvalues. In particular, we focus on a set of eigenfunctions represented by
formulas with generalized Laguerre functions. This set of eigenfunctions can be
written as linear combinations of two subsets of eigenfunctions, one of which
is closed under the star-multiplication with the creation operator to a generic
complex power -- and the other similarly with the annihilation operator. The
two subsets intersect when the left and the right eigenvalues differ by an
integer. We further investigate how star-multiplications with both the creation
and annihilation operators together may change such eigenfunctions and briefly
discuss some problems that we are facing in order to use these eigenfunctions
as the initial data to construct solutions to Vasiliev's equations.
