  Iridates supply fertile grounds for unconventional phenomena and exotic
electronic phases. With respect to well-studied octahedrally-coordinated
iridates, we pay our attention to a rather unexplored iridate, Na4IrO4, showing
an unusual square-planar coordination. The latter is key to rationalize the
electronic structure and magnetic property of Na4IrO4, which is here explored
by first-principles density functional theory and Monte Carlo simulations. Due
to the uncommon square-planar crystal field, Ir 5d states adopt
intermediate-spin state with double occupation of dz2 orbital, leading to a
sizable local spin moment, at variance with many other iridates. The
square-planar crystal field splitting is also crucial in opening a robust
insulating gap in Na4IrO4, irrespective of the specific magnetic ordering or
treatment of electronic correlations. Spin-orbit coupling plays a minor role in
shaping the electronic structure, but leads to a strong magnetocrystalline
anisotropy. The easy axis perpendicular to the IrO4 plaquette, well explained
using perturbation theory, is again closely related to the square-planar
coordination. Finally, the large single-ion anisotropy suppresses the spin
frustration and stabilizes a collinear antiferromagnetic long-range magnetic
ordering, as confirmed by Monte Carlo simulations predicting a quite low N\'eel
temperature, expected from almost isolated IrO4 square-planar units as
crystalline building blocks.
