  In this work, we extend previous results, demonstrating how complexity in an
open quantum system can identify decoherence between two fields, even in the
presence of an accelerating background. Using the curved-space Caldeira-Leggett
two-field model in de Sitter as our toy model, we discover a distinctive
feature in the growth of complexity of purification, providing an alternative
diagnostic for studying decoherence when the adiabatic perturbation is coupled
to a heavy field. This paper initiates a new pathway to explore the features of
quantum complexity in an accelerating background, thereby expanding our
understanding of the evolution of primordial cosmological perturbations in the
early universe.
