  We investigate relativistic quantum mechanics (RQM) for particles with
arbitrary magnetic moment. We compare two well known RQM models: a) Dirac
equation supplemented with an incremental Pauli term (DP); b) Klein-Gordon
equations with full Pauli EM dipole moment term (KGP). We compare exact
solutions to the external field cases in the limit of weak and strong
(critical) fields for: i) homogeneous magnetic field, and ii) the Coulomb
$1/r$-potential. For i) we consider the Landau energies and the Landau states
as a function of the gyromagnetic factor ($g$-factor). For ii) we investigate
contribution to the Lamb shift and the fine structure splitting. For both we
address the limit of strong binding and show that these two formulations
grossly disagree. We discuss possible experiments capable of distinguishing
between KGP and DP models in laboratory. We describe impact of our
considerations in astrophysical context (magnetars). We introduce novel RQM
models of magnetic moments which can be further explored.
