In this work, we investigate the physical aspects of the inflection-point inflation scenario and assess its observational viability in light of current cosmic microwave background (CMB) data. The model we consider encapsulates the inflaton with a pseudoscalar (the dark matter candidate) in a complex neutral scalar singlet. The cosmological constraints on the parameters of inflation derived at a high-energy scale are translated to a low-energy scale by running these parameters. Ensuring the entire Lagrangian to be invariant under a symmetry with the adequate transformation of the fields, the imaginary part of the singlet decouples from the other scalars of the model. We then investigate if the observational viability of inflation is also compatible with this pseudoscalar being the dark matter component. We show that the CMB constraints on the inflationary parameters assure that the pseudoscalar is stable and provides the correct relic dark matter abundance only when the pseudoscalar is thermally produced.