Planets experience numerous large impacts at the end of the planetary accretion stage. Such large impacts can produce deep magma oceans (MOs) within planets. During magma ocean crystallization, part of the magma ocean can become enriched in iron and therefore form a dense melt that migrates downward to create an iron-enriched basal magma ocean (BMO) at the core–mantle boundary. These MOs and BMOs can generate dynamos if they are convectively unstable and if their electrical conductivity is sufficiently high. This may have occurred in the early Earth and can occur in super-Earths, but it requires very high electrical conductivity of these magma oceans, which is not yet fully constrained by experiments. To investigate the likelihood of MO- and BMO-driven dynamos, we estimate the electrical conductivity of MO- and BMO-analogue materials by conducting laser-driven shock experiments, density functional theory molecular dynamics simulations (DFT-MD), and thermal and magnetic field evolution modeling of Earth and super-Earths. We find that super-Earths larger than 3–6 Earth masses likely produce BMO-driven dynamos lasting a few billion years, which may be observable in the future. The early Earth could also have hosted a strong BMO-driven dynamo if the BMO electrical conductivity was sufficiently high.