This paper proposes a novel nature-inspired $α$-fair hybrid precoding (NI-$α$HP) technique for millimeter-wave multi-user massive multiple-input multiple-output systems. Unlike the existing HP literature, we propose to apply $α$-fairness for maintaining various fairness expectations (e.g., sum-rate maximization, proportional fairness, max-min fairness, etc.). After developing the analog RF beamformer via slow time-varying angular information, the digital baseband (BB) precoder is designed via the reduced-dimensional effective channel matrix seen from the BB-stage. For the $α$-fairness, we derive the optimal digital BB precoder expression with a set of parameters, where optimizing them is an NP-hard problem. Hence, we efficiently optimize the parameters in the digital BB precoder via five nature-inspired intelligent algorithms. Numerical results present that when the sum-rate maximization is the target, the proposed NI-$α$HP technique greatly improves the sum-rate capacity and energy-efficiency performance compared to other benchmarks. Moreover, NI-$α$HP supports different fairness expectations and reduces the rate gap among UEs by varying the fairness level ($α$).