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Exploring the effect of exotic fields around black holes on particle dynamics may help to understand the nature of dark matter and energy. The quintessential field can be treated as one of such fields. In this work, we investigate the motion of spinning particles in the vicinity of rotating black holes immersed in quintessential dark energy, characterized by the equation of state (EoS) parameter omega is an element of (-1; -1/3) governing the equation of state of the dark energy and dimensionless quintessential field parameter C. Using the Mathisson-Papapetrou- Dixon (MPD) equations, we derive the effective potential and study superluminal bound values for the particle spin. Also, we investigate the behaviors of the innermost and outermost stable circular orbit (ISCO & OSCO) of spinning test particles and their energy and angular momentum at the orbits. Note that the OSCO exists due to the third cosmological-like horizon caused by the quintessential field and shows that the ISCO and OSCO coincide at critical values of the quintessential field and EoS parameters, which also depend on the particle and black hole spin. Finally, we explore collisions of spinning particles and analyze the center-of-mass energies and critical angular momentum, which allows the collisions of the particles near the black hole.