Self-propulsion of magneto-elastic composite microswimmers was demonstrated under a uniaxial field at both the air-water and the water-substrate interfaces. The microswimmers were made of elastically-linked magnetically hard CoNiP and soft Co ferromagnets, fabricated using standard photolithography and electrodeposition. Swimming speed and direction were dependent on the field frequency and amplitude, reaching a maximum of 95.1 µm/s on the substrate surface. Fastest motion occurred at low frequencies, via a spinning (air-water interface) or tumbling (water-substrate interface) mode that induced transient inertial motion. Higher frequencies resulted in low Reynolds number propagation at both interfaces via a rocking mode. Therefore the same microswimmer could be operated as either a high or a low Reynolds number swimmer. Swimmer pairs agglomerated to form a faster superstructure that propelled via spinning and rocking modes analogous to those seen in isolated swimmers. Microswimmer propulsion was driven by a combination of dipolar interactions between the Co and CoNiP 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.