We discuss a complex singlet extension of the Standard Model (SM) with an extra Z_2 symmetry on the imaginary part, which can provide a dark matter candidate and explain the baryon asymmetry of the Universe. We analyze the the Z_2 broken and Z_2 preserving (or dark-matter) phases and perform a theoretical study of the renormalization group (RG), deriving the two-loop beta functions for the scalar sector. We also discuss RG stability and find that for both models, a 126 GeV Higgs boson mass combined with stability up to, at least, an intermediate scale (M~10^10 GeV), predicts a new scalar state with a lower bound of 140 GeV (the same conclusion applies for the heaviest scalar in the broken phase with no dark matter). We include full phenomenological studies imposing bounds and measured signal rates for the Higgs boson at the end of the 8 TeV run of the LHC as well as dark matter constraints. Combining both theoretical results with the experimental bounds, we find complete model points which saturate the dark-matter relic density, are consistent with experimental data and are stable up to the GUT scale, providing a new scalar which may be observable at the 13/14 TeV LHC runs.