  In the present work, we intend to predict the production rates of the Higgs
bosons in the simplest extension of the Standard Model (SM) by a neutral
complex singlet (cSMCS). This model has an additional source of CP violation
and provides strong enough first-order electroweak phase transition to generate
the baryon asymmetry of universe (BAU). The scalar spectrum of the cSMCS
includes three neutral Higgs particles with the lightest one considered to be
the 125 GeV Higgs boson found at LHC. The SM-like Higgs boson comes mostly from
the SM-like SU(2) doublet, with a small correction from the singlet. To predict
the production rates of the Higgs bosons, we use a conventional effective LO
QCD framework and the unintegrated parton distribution functions (UPDF) of
Kimber-Martin-Ryskin (KMR). We first compute the SM Higgs production
cross-section and compare the results to the existing theoretical calculations
from different frameworks as well as the experimental data from the CMS and
ATLAS collaborations. It is shown that our framework is capable of producing
sound predictions for these high-energy QCD events in the SM. Afterwards we
present our predictions for the Higgs boson production in the cSMCS.
