The nature of dark matter is a fundamental problem in cosmology and particle physics. Many particle candidates have been devised over the course of the last decades, and are still at stake to be soon discovered or rejected. However, astronomical observations, in conjunction with the phenomenological efforts in astrophysical modeling, as well as in particle theories to explain them, have helped to pin down several key properties which any successful candidate has to have. In this talk, I will explore the possibility that dark matter is described by a complex scalar field (SFDM), while the other cosmic components are treated in the usual way, assuming a cosmological constant for the dark energy. We will see that the background evolution of a Universe with SFDM and a cosmological constant (LSFDM) complies with the concordance LCDM model, if the model parameters of the SFDM Lagrangian, mass and repulsive 2-particle self-interaction coupling strength, are properly constrained by observations of the cosmic microwave background and Big Bang nucleosynthesis (BBN). However, not only does LSFDM lead to non-standard expansion histories prior to BBN, it also exemplifies differences at small scales, which could help to resolve the discrepancies found between LCDM and certain galaxy observations. I will highlight the differences between complex SFDM and dark matter described by real fields, as for instance axion-like particles. If time permits, I will also talk about possible implementations of SFDM in the very early Universe, in the wake of its inflationary phase.