We propose a quantum-inspired framework for modeling open distributed intelligence systems (DISs) comprising natural intelligence agents (NIAs) and artificial intelligence agents (AIAs) that interact with each other. Each NIA - AIA pair represents a user and their digital assistant - an avatar implemented as an agent based on a large language model (LLM). The AIAs are interconnected through a complex, scale-free network and communicate with users and one another in real time. We focus on the social impact and evolution of users' emotional states, which we model as simple, two-level cognitive systems shaped by interactions with AIAs and external information sources. Within this framework, the AIAs adiabatically follow the NIAs, mediating emotional influence by disseminating information and propagating user emotions throughout the system. Building on Mehrabian's Pleasure-Arousal-Dominance (PAD) model and Wundt's three-dimensional theory of emotions, we put forward a quantum-like representation of affective states on an emotional sphere. We demonstrate that the arousal component is governed by the interplay between external informational inputs and individual personality traits. This leads to the emergence of limiting cycles in emotional dynamics. Assuming weak AIA - AIA coupling, we identify two distinct regimes of affective behavior. In the first regime, coherent NIA - AIA interaction supports emotional heterogeneity and individual differentiation across the network. In the second regime, shared exposure to external information drives synchronized emotional responses, resulting in a macroscopic affective field that captures collective emotional dynamics. Furthermore, we demonstrate that the network's structural properties, particularly node degree correlations, play a role analogous to quantum correlations in ensembles of two-level physical systems; a quantum-like superradiant state corresponds to the network-induced collective emotional activation of NIAs within a DIS. These findings advance our understanding of affective dynamics and emergent social phenomena in hybrid human-AI ecosystems.