There is an alternative way of introducing nonlinearity into QM, at the level of QFT, that your previous question did not admit. QFT is usually presented as a theory in terms of a quantum field such as $\hat\phi(x)$, but this is not an operator, it is an operator-valued distribution, which lets us construct operators by "smearing" with "test functions" (which in signal processing would be called "window functions", which I believe may be the best way for non-mathematicians to start to understand this formalism), $$\hat\phi:\mathcal{S}\rightarrow\mathcal{A};f\mapsto\hat\phi_f=\int\hat\phi(x)f(x)\mathrm{d}^4x.$$
$\hat\phi$ is a linear map from functions on space-time to operators, $\hat\phi_f$. We can construct self-adjoint observables using sums of products of operators $\hat\phi_{f_i}$, whereas, again, $\hat\phi(x)$ is not an operator.
To obtain a probability theory, we have to introduce a state over the $\star$-algebra of observables that is generated by the $\hat\phi_f$, for whatever test functions we choose to use. The state has to be a normalized, complex-linear, positive semi-definite map (which means that $\omega(1)=1$, $\omega(\lambda\hat A)=\lambda\omega(\hat A)$, $\omega(\hat A+\hat B)=\omega(\hat A)+\omega(\hat B)$, and $\omega(\hat A^\dagger\hat A)\ge 0$). That's enough for us to construct a Hilbert space (using what's called the GNS construction), and then we can introduce a Born probability interpretation. The elementary way to construct a state is to introduce a vacuum vector $\left|0\right>$ and the trivial action of annihilation operators on it, $a_f\left|0\right>=0$, and then to define $\hat\phi_f=a_{f^*}+a_f^\dagger$ and the vacuum state as $\omega(\hat A)=\left<0\right|\hat A\left|0\right>$, which, with the commutation relation between the creation and annihilation operators, gives you the vacuum sector of all the quantized free fields (but many other states can be constructed). Note, however, that although the state has to be linear over the algebra for us to be able to construct a probability interpretation, there is nothing that requires that the map from functions on space-time to the algebra of operators has to be linear. That is, we can construct a Hilbert space and a probability interpretation even if $\hat\phi:\mathcal{S}\rightarrow\mathcal{A};f\mapsto\hat\phi_f$ is nonlinear.
No reason is given, but the Wightman axioms insist that the quantum field must be a linear map from functions on space-time to the algebra of operators, and there is a more abstract insistence on something very similar in the Haag-Kastler axioms, so this option is essentially unconsidered in the literature. It opens up a different world if we consider them. Note that this is not so much a theory beneath QM, it is essentially about the relationship between whatever abstract operators we use in QM and whatever we take to correspond to them in space-time (that's sort of beneath, but is more an intrinsic question that has to be answered by any abstract approach to QM, such as the Aaronson paper that was mentioned in an Answer to your previous Question, which is entirely right as far as it goes).
AFAIK, the above has essentially no connection with quantum gravity, noncommutative space-times, or any of the more popular research directions to be found in the literature today.
