Polysaccharide-based hydrogels have gained significant attention in tissue engineering because of their biocompatibility, biodegradability, and ability to mimic extracellular matrix conditions. Alginate is widely explored for regenerative applications; however, its limited mechanical stability and lack of specific cell-interaction sites may restrict its performance. The incorporation of chitosan together with chemical crosslinking represents an effective strategy to improve the structural and physicochemical properties of these biomaterials. This work aimed to develop hybrid alginate–chitosan hydrogel networks crosslinked with polyamine-based crosslinkers as potential platforms for tissue regeneration. Alginate–chitosan hydrogels were synthesized through polymeric network formation and chemically crosslinked using 1,4-diaminobutane, piperazine, and spermidine. The obtained materials were characterized by Fourier-transform infrared spectroscopy (FT-IR), electrospray ionization mass spectrometry (ESI-MS), scanning electron microscopy (SEM), rheological analysis, swelling behavior, degradation assays, and in vitro cytocompatibility evaluation using fibroblast cells. Spectroscopic analyses confirmed the formation of crosslinked hydrogel networks through amide bond formation. SEM revealed interconnected porous three-dimensional structures whose morphology depended on the polymer composition and crosslinking system. The incorporation of chitosan increased the structural integrity of the hydrogels, allowing reliable rheological characterization. Among the evaluated formulations, AAQ0.7 exhibited the best overall performance, showing predominant elastic behavior (G' > G''), the highest swelling capacity (162.92%), and the lowest degradation rate under acidic conditions compared with the other formulations. Cytocompatibility assays indicated that the alginate–chitosan hydrogels were non-cytotoxic to fibroblasts, although limited cell adhesion and proliferation were observed. These preliminary findings suggest that hybrid alginate–chitosan hydrogels crosslinked with polyamine-based crosslinkers are promising candidates for further development as scaffolds for tissue engineering applications.