  We propose a minimal toy model for a two-dimensional altermagnet. The model
unravels altermagnetic properties at a microscopic level. We find spin-split
electron- and non-degenerate magnon bands with a $d$-wave symmetry. We use the
model to explore magnon-mediated superconductivity in altermagnets. The
dominant superconducting state is spin-polarized with a $p$-wave symmetry. The
state adopts its characteristics from the spin-split electron bands.
Furthermore, we find that the superconducting critical temperature of
altermagnets can be significantly enhanced by tuning the chemical potential.
