Tackling the increasing energy demand of our society is one of the key challenges today. With the rise of artificial intelligence, information and communication technologies started to substantially contribute to this alarming trend and therefore necessitate more sustainable approaches for the future. Brain-inspired computing paradigms represent a radically new and potentially more energy-efficient approach for computing that may complement or even replace CMOS in the long term. In this perspective, we elaborate on the concepts and properties of complex chemical reaction networks (CRNs) that may serve as information-processing units based on chemical reactions. The computational capabilities of simpler, oscillatory chemical reactions have already been demonstrated in scenarios ranging from the emulation of Boolean gates to image-processing tasks. CRNs offer higher complexity and larger non-linearity, potentially at lower energy consumption. Key challenges for the successful development of CRN-based computers are associated with their specific physical implementations, operability, and readout modalities. CRNs are sensible to various reaction triggers, and provide multiple and interlinked reaction pathways and a diverse compound space. This bears a high potential to build radically new hardware and software concepts for energy-efficient computing based on neuromorphic architectures—with computing capabilities in real-world applications yet to be demonstrated.