Neural tissue engineering (TE) represents a promising new avenue of therapy to support nerve recovery and regeneration. To recreate the complex environment in which neurons develop and mature, the ideal biomaterials for neural TE require a number of properties and capabilities including the appropriate biochemical and physical cues to adsorb and release specific growth factors. Here, we present neural TE constructs based on electrospun serum albumin (SA) fibrous scaffolds. We doped our SA scaffolds with an iron-containing porphyrin, hemin, to confer conductivity, and then functionalized them with different recombinant proteins and growth factors to ensure cell attachment and proliferation. We demonstrated the potential for these constructs combining topographical, biochemical, and electrical stimuli by testing them with clinically relevant neural populations derived from human induced pluripotent stem cells (hiPSCs). Our scaffolds could support the attachment, proliferation, and neuronal differentiation of hiPSC-derived neural stem cells (NSCs), and were also able to incorporate active growth factors and release them over time, which modified the behavior of cultured cells and substituted the need for growth factor supplementation by media change. Electrical stimulation on the doped SA scaffold positively influenced the maturation of neuronal populations, with neurons exhibiting more branched neurites compared to controls. Through promotion of cell proliferation, differentiation, and neurite branching of hiPSC-derived NSCs, these conductive SA fibrous scaffolds are of broad application in nerve regeneration strategies.