  In this paper, by considering the complex form of the quintessence model, we
study two different dynamic structures of holographic dark energy as Tsallis
and Kaniadakis in the framework Brans-Dicke cosmology. In each setup, we employ
non-interacting and interacting cases and calculate some cosmological
parameters such as the equation of state $\omega$. We also discuss the $
\omega- \omega '$ behavior. By modifying the potential and studying the scalar
field dynamics, we examine the complex quintessence cosmology.
  In addition, considering the two parts of the quintessence field effects,
i.e., real and complex, and considering the fractional energy density $
\Omega_{D} $, we examine whether it can describe a real universe or not.
  We also specify that the fractional energy density can not be arbitrary
between 0 and 1. In other words, it depends on the Tsallis, Kaniadakis, and
Brans-Dicke cosmology free parameters.
  We create a relationship between the fractional energy density and other
parameters introduced in the text such as $\delta$, $b^{2}$, $\alpha$ and
$\beta$ for each model separately. Finally, we compare the obtained results of
models to each other and the latest observable data.
