In this work we explore the possibility of incorporating particle physics motivated scalar fields to the dark matter cosmological model. In this landscape, we consider the classical complex scalar field in a certain region in the parameter space of the model that increases the number of neutrino species Neff, in order to be consistent with the observed abundance of light elements produced at Big Bang Nucleosynthesis (BBN). We perform analyses using one and two scalar fields. In the two scalar field models we explore the possibility of combining an axion and a Higgs-like field as well as a Higgs-like field and the classical field, we show that in the first case there is no set of parameters that allows us to be consistent with Neff, while in the second case a strong restriction to the set of parameters is obtained. This last restriction is given in terms of a maximum bound of the fraction of Higgs-like field that can be incorporated together with the classical field. Our results could be relevant in the direct dark matter detection programs.