Researchers have developed a new metric describing a rotating cosmological black hole, characterized by possessing electric and magnetic charges (dyonic), and immersed in a quintessence-type dark energy environment. This model incorporates both the cosmological constant and quintessence as components of external matter influencing spacetime geometry. The construction starts from a Schwarzschild-type metric, incorporating all energy contributions and applying the Newman-Janis algorithm with a unique and general complexification rule to introduce rotation, thus avoiding ambiguities in the formulation.

The study focuses on verifying that this metric satisfies Einstein's field equations in the presence of matter. Conditions for the event horizon, the total stress-energy tensor ($T_{\mu\nu}$), and the Kretschmann and Ricci curvature scalars have been calculated. A key aspect of the research is the analysis of how the cosmological constant $\Lambda$ and the quintessence parameter $\alpha$ affect the position of the event horizon and the ergosphere of the black hole. It is observed that in the limit $\alpha \rightarrow 0$, the scalar curvature differs from the vacuum result ($R \neq -4\Lambda$), while the singularity region remains unchanged.

Finally, the work explores the dynamical properties of test particles in this environment. The angular and rotational velocities of a test particle are investigated, as well as the energy conditions (energy density and pressures) required for the existence and stability of this black hole solution. This model provides a theoretical tool to better understand the interaction between complex black holes and dark energy in a cosmological context.