  The Cram\'er-Rao, Fisher-Shannon and LMC shape complexity measures have been
recently shown to play a relevant role to study the internal disorder of finite
many-body systems (e.g., atoms, molecules, nuclei). They highlight amongst the
bunch of complexities appeared in the literature by their mathematical
simplicity and transparent physical interpretation. They are composed by two
spreading measures (variance, Fisher information and Shannon entropies) of the
single-particle probability density of the systems. Here we discuss the
physico-mathematical knowledge of these two-component complexities for
hydrogenic systems with standard ($D = 3$) and non-standard dimensionalities.
These systems include not only the hydrogen atom and its isoelectronic series
but also a large diversity of quantum systems such as, e.g. exotic atoms,
antimatter atoms, some qubits systems and Rydberg atoms, ions and molecules.
