Self-propulsion of magneto-elastic composite microswimmers is demonstrated under a uniaxial field at both the air-water and the water-substrate interfaces. The microswimmers are made of elastically linked magnetically hard Co-Ni-P and soft Co ferromagnets, fabricated using standard photolithography and electrodeposition. Swimming speed and direction are dependent on the field frequency and amplitude, reaching a maximum of 95.1 μm/s on the substrate surface. Fastest motion occurs at low frequencies via a spinning (air-water interface) or tumbling (water-substrate interface) mode that induces transient inertial motion. Higher frequencies result in low Reynolds number propagation at both interfaces via a rocking mode. Therefore, the same microswimmer can be operated as either a high or a low Reynolds number swimmer. Swimmer pairs agglomerate to form a faster superstructure that propels via spinning and rocking modes analogous to those seen in isolated swimmers. Microswimmer propulsion is driven by a combination of dipolar interactions between the Co and Co-Ni-P magnets and rotational torque due to the applied field, combined with elastic deformation and hydrodynamic interactions between different parts of the swimmer, in agreement with previous models.