The first question popping up is where we might begin looking for this hypothetical language of Quantum Physics. One is naturally lead to think that a departing point is the Bohr's correspondence principle, who originated the quantum theory by trying to reveal its mathematical substrate.
The Bohr's correspondence principle—accordingly to which Classical Physics has to be treated as a limit case of the Quantum one—was the first heuristic method leading to the construction of mathematical models for quantum phenomena. Taking into account that the natural language for Classical Physics is Differential Calculus, it may be assumed that Differential Calculus is a limit case—or, better to say, a degenerate case—of a more general mathematical theory going under the name of "Quantistic" Differential Calculus. This general idea represents the mathematical paraphrase of the correspondence principle, the mathematical correspondence principle. This principle lead us to think that the mathematical language of Quantum Physics has to be a natural extension of the Differential Calculus, instead of a chaos of appendages and superstructures formed by many constructions of Functional Analysis, Measure Theory, Non-commutative Algebra, and so on.
The mathematical correspondence principle is not just a mere indication of the existence of a mysterious Quantistic Differential Calculus. Indeed it can be notice that the ordinary differential equations of Classical Mechanics describe the behavior of the singularities of the solutions of the differential equations of the Quantum Mechanics. That are equations in partial derivatives. On the other hand, the departing point for the theory of quantum fields is the equations of classical fields, that by analogy describe the behavior of the singularities of the quantum fields. But classical fields are described by means of partial differential equations, so that—assuming the validity of the correspondence principle in this situation as well—we are brought to the conclusion that the equations describing quantum fields undoubtedly have to be differential equations of an unknown first genre being in the same relationship with partial differential equations as the partial differential equations are with the ordinary ones. This way the mathematical correspondence principle not only tells about the existence of the Quantum Differential Calculus, but also states that it must be looked for within a duly developed theory of partial differential equations.
Historically Bohr's principle has been used only for technical purposes, allowing to guess some quantization schemes. Its hidden mathematical aspect can be enlightened and comprehended in the era of formation of the existing quantum theory. One reason for this was the marginal state of the studies of the non linear PDEs, while everybody believed—accordingly to current trends—that truth was hidden into Hilbert spaces. Therefore von Neumann's statement that theories of self-adjoint operators on Hilbert spaces would represent the mathematical foundations of Quantum Mechanics was enthusiastically embraced by the wide public in spite of the fact that it violated Bohr's correspondence principle in a shocking rough way. It has been rumored that only Levi-Civita tried very timidly to display his own disagreement, but his subdued voice was not heeded. It was an "ecological catastrophe", whose consequences would live on for a long time.