Sadly, with lorentz invariance things get quite a bit more complicated. Adding in Lorentz invariance forces you to deal more directly with spin (and lets you prove spin-statisics), so you end up with the Klein-Gordon equation for spin 0, the Dirac equation for spin 1⁄2, and variants of the Maxwell equations for spin 1.
But you also get weird “paradoxical” effects trying to interpret the results of those equations along the lines of non-relativistic quantum, so you are forced to push towards full field theory.
Sadly, with lorentz invariance things get quite a bit more complicated. Adding in Lorentz invariance forces you to deal more directly with spin (and lets you prove spin-statisics), so you end up with the Klein-Gordon equation for spin 0, the Dirac equation for spin 1⁄2, and variants of the Maxwell equations for spin 1.
But you also get weird “paradoxical” effects trying to interpret the results of those equations along the lines of non-relativistic quantum, so you are forced to push towards full field theory.