Bispecific antibodies (BsAbs) can bind to two antigens simultaneously and have undergone rapid advancements in recent years owing to their ability to enable novel mechanisms of action that are unachievable using conventional monoclonal antibodies (mAbs). However, the structural complexity of BsAbs remains a problem during product development. One of these problems is the presence of impurities and by-products. Although BsAbs with the human Fc region must be assembled using heterogeneous polypeptide chains, undesired by-products from unpaired and mispaired chain components can contaminate them. These by-products are difficult to remove in the purification process because their physicochemical properties resemble those of the target BsAb with correct pairing. Here, we designed a novel Fc region for enhanced BsAbs in which the human CH2 domain on one side of the Fc region was replaced with the C2 domain from an immunoglobulin new antigen receptor (IgNAR) shark antibody. The designed BsAbs with chimeric and asymmetric Fcs exhibited separate pH elution profiles against soluble aggregates in protein A affinity chromatography. An overlapping elution profile corresponding to the by-product homogeneous chain observed in human Fc BsAbs was not detected in shark C2-introduced BsAbs. Although another homogeneous by-product was observed in the designed BsAb, introducing N-glycosylation at C2 significantly improved this problem. Additionally, BsAbs with the designed Fc demonstrated higher stability in both the colloidal and structural aspects. This study is the first approach for the chimeric and asymmetric design of Fc using a shark-derived constant domain and offers a novel alternative for BsAb development.