In this study, a square-root-complex fuzzy set (SR-CFS) frame is introduced to associate Lie algebraic structures for modeling interaction-aware hesitancy in multi-criteria decision making processes. Instead of using real-valued memberships as in classical fuzzy extensions, our method deals with squared and square-rooted complex membership values and phase elements simultaneously to show the strength of evidence, uncertainty, and disagreement among experts. In this structure, we define SR-complex fuzzy Lie subalgebras (SR-CFLSAs) and SR-complex fuzzy Lie ideals (SR-CFLIDs) and illustrate their basic properties. They consist of stability under Lie algebra homomorphisms, which ensure that the structure of uncertainty is preserved during algebraic operations. Based on these theories, we combine SR-CFS with the PROMETHEE Ⅱ and VIKOR methods to develop an integrated decision-making system to deal with interdependent criteria effectively. The work includes an extensive case study on AI-assisted diagnosis of newborn tuberculosis, where we derive decision matrices from expert evaluations and clinical performance metrics.