  We analyze theoretically two different setups for s-wave superconductivity
(SC) proximitized $j=3/2$ particles in Luttinger materials that are able to
host Majorana bound states (MBSs). First, we consider a one-dimensional SC wire
with intrinsic bulk inversion asymmetry (BIA). In contrast to wires, modeled by
a quadratic dispersion with Rashba spin-orbit coupling, there are two
topological phase transitions in our systems at finite magnetic fields. Second,
we analyze a two-dimensional Josephson junction on the Luttinger model finding
a topological region even in the absence of BIA and Rashba spin-orbit
couplings. This originates from the hybridization of the light and heavy hole
bands of the $j=3/2$ states in combination with the SC pairing. As a
consequence, both systems can be driven into a topological phase hosting MBSs.
Hence, we predict that MBSs form in any SC proximitized Josephson junction on
2D Luttinger materials by the application of magnetic field alone. This opens a
new avenue for the search of topological SC.
