It is shown in the Weyl limit-point case that system of root functions of the non-self-adjoint Bessel operator and its perturbation Sturm-Liouville operator form a complete system in the Hilbert space. Furthermore, asymptotic behavior of the eigenvalues of the non-self-adjoint Bessel operators is investigated, and it is proved that system of root functions form a Bari basis in the same Hilbert space. Copyright (c) 2013 John Wiley & Sons, Ltd.