This paper explores the relationship between quantum complexity and decoherence in an open quantum system, specifically in the context of the early universe. The authors use the curved-space Caldeira-Leggett model in de Sitter space as a toy model to investigate how complexity can be used as a diagnostic tool for studying decoherence. They focus on the interaction between a light field (representing primordial cosmological perturbations) and a heavy field (representing an entropic degree of freedom). By analyzing the time evolution of complexity, particularly the complexity of purification (COP), they find that a distinctive feature in the growth of complexity provides a signature of decoherence. This signature is particularly prominent in the regime where the coupling strength is less than the critical value \(3MH/2\), where a bump in the COP growth rate aligns with the saturation timescale of linear entropy, a measure of decoherence. The authors suggest that this bump could serve as a reliable indicator of decoherence in more realistic models of inflation, providing a new pathway to explore quantum complexity in accelerating backgrounds.This paper explores the relationship between quantum complexity and decoherence in an open quantum system, specifically in the context of the early universe. The authors use the curved-space Caldeira-Leggett model in de Sitter space as a toy model to investigate how complexity can be used as a diagnostic tool for studying decoherence. They focus on the interaction between a light field (representing primordial cosmological perturbations) and a heavy field (representing an entropic degree of freedom). By analyzing the time evolution of complexity, particularly the complexity of purification (COP), they find that a distinctive feature in the growth of complexity provides a signature of decoherence. This signature is particularly prominent in the regime where the coupling strength is less than the critical value \(3MH/2\), where a bump in the COP growth rate aligns with the saturation timescale of linear entropy, a measure of decoherence. The authors suggest that this bump could serve as a reliable indicator of decoherence in more realistic models of inflation, providing a new pathway to explore quantum complexity in accelerating backgrounds.