Complex quintessence theory, Tsallis and Kaniadakis holographic dark energy and Brans–Dicke cosmology

In this paper, we study two different dynamic structures of holographic dark energy, namely Tsallis and Kaniadakis, within the framework of Brans–Dicke cosmology. We consider the complex form of the quintessence model and examine both non-interacting and interacting cases, calculating various cosmological parameters such as the equation of state [Formula: see text] and discussing the behavior of [Formula: see text]. We modify the potential and study the scalar field dynamics of complex quintessence cosmology. Additionally, we examine the effects of the two parts of the quintessence field (real and complex) and the fractional energy density [Formula: see text], determining whether they can describe a real universe. We note that the fractional energy density cannot be arbitrary between 0 and 1, as it depends on the Tsallis, Kaniadakis and Brans–Dicke cosmology-free parameters. For each model, we establish a relationship between the fractional energy density and other parameters such as [Formula: see text], [Formula: see text], [Formula: see text] and [Formula: see text].