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Recently, we have introduced in [A. M. Marques et al., Phys. Rev. B 103,235425 (2021)] the concept of $2^n$-root topology and applied it toone-dimensional systems. These models require $n$ squaring operations to theirHamiltonians, intercalated with different constant energy downshifts at eachlevel, in order to arrive at a decoupled block corresponding to a knowntopological insulator (TI) that acts as the source of the topological featuresof the starting $2^n$-root TI ($\sqrt[2^n]{\text{TI}}$). In the process, $n$non-topological residual models with degenerate spectra and in-gap impuritystates appear, which dilute the topologically protected component of thestarting edge states. Here, we generalize this method to severaltwo-dimensional models, by finding the 4-root version of lattices hosting weakand higher-order boundary modes (both topological and non-topological) of aChern insulator and of a topological semimetal. We further show that a startingmodel with a non-Hermitian region in parameter space and a complex energyspectrum can nevertheless display a purely real spectrum for all its successivesquared versions, allowing for an exact mapping between certain non-Hermitianmodels and their Hermitian lower root-degree counterparts. A comment is made onthe possible realization of these models in artificial lattices.

$2^n$-root weak, Chern, and higher-order topological insulators, and $2^n$-root topological semimetals