There are several attempts to build an alternative to quantum mechanics in the sense you are asking (i.e. including generalizations). Some of those attempts are rather advanced both in its formulation and understanding of the approach; others attempts are only available in a sketchy form and probably will never succeed. Some early attempts were found to disagree with experiments and then almost abandoned, but latter they were improved to solve the deficiencies and are being considered again as alternatives to quantum theory; this is the case with stochastic electrodynamics (SED), of which you can find several versions of them in the literature, with the former versions abandoned even by its practitioners.
Basically you can find two main approaches to alternatives to quantum mechanics in the sense you are asking: the first approach consists on attempts that consider that quantum theory is a fundamental theory and maintain its mains elements, but modify some that is perceived as a flaw. This is the case with the irreversible extensions of quantum mechanics, where the Schrödinger equation of motion is replaced by an irreversible equation of motion, but the rest of the quantum formalism is maintained intact: operators, matrices,... The work of the Brussels-Austin school or the work of the Keenan school at MIT are two examples of attempts to formulate an irreversible extension of quantum mechanics. Some Solvay Conference on Physics have been devoted to this topic. You can check the different conference contributions at next link
Keenan's school work can be found in next page and in the links cited therein
Very related to those approaches are the theories with objective collapse, because collapse is modeled as an irreversible process and requires an extension of the Schrödinger equation: check Ghirardi–Rimini–Weber theory and Penrose quantum gravity formulation, for instance.
The second approach to alternatives to quantum mechanics includes all those attempts to formulate a completely different theory: e.g., a deterministic theory without wavefunctions, matrices, and operators. Those attempts consider that quantum mechanics is not fundamental but emergent and valid only under certain limits. Most of those attempts are characterized by reintroducing some aspects of classical physics abandoned by quantum mechanics, but do not make the mistake of believing this is a return to pre-quantum classical physics. At contrary, those novel theories are better considered "neoclassical" or "superclassical" in the sense that they can explain phenomena that cannot be explained by classical physics such as double-slit experiment.
The above mentioned SED belongs to this class. Next link gives a discussion of recent advances in Pilot-Wave Hydrodynamics and its relationship with deBroglie approach and SED
Other developments to emergent quantum mechanics include Sachs' Theory of Inertia or Adler's trace dynamics. There is a set of conferences devoted to the topic of emergent quantum mechanics, exploring the derivation of quantum theory from something more fundamental. Papers for second and third conference can be freely accessed next