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We identify $mathbb{R}^4$ with the quaternions $mathbb{H}={t=x+yi+zj+wkmid x,y,z,win mathbb{R}}$. We define the differential operator $D$ on $C^{infty}(mathbb{R}^4)$, the space of smooth quaternion-valued maps, via
$$
D(f):= frac{partial}{partialbar{t}}f,
$$
where notice that $partial/partialbar{t} = partial/partial x+ ipartial /partial y +jpartial/partial z+kpartial /partial w$.

Is $ker D$ closed under the quaternionic multiplication? Is this kernel closed under the topology of uniform convergence on compact subsets? Does this kernel contain a non-constant bounded function? (A kind of Lioville theorem?)

Is there a terminology and a classification of all $4$-manifolds with an atlas with the property that this operator can be well defined independently of the choice of a local chart?

I vaguely remember that the generalization of holomorphic calculus in the quaternion setting has some technical difficulties if we generalize differentiability as existence of the usual limit $lim_{tto t_0} frac{f(t)-f(t_0)}{t-t_0}$.

Is the above elliptic differential operator $D$ an appropriate remedy for such technical difficulty?

Quaternionic analysis is less well behaved than complex analysis. Defining the functions spaces through kernel of appropriate generalization of Cauchy-Riemann operator leads to Clifford analysis which generalizes these question to yet broader setting. The quaternionic case was studied by Fueter in 1930s. For questions about function spaces I suggest you check out Clifford Algebra and Spinor-Valued Functions: A Function Theory for the Dirac Operator by Delanghe, R., Sommen, F., Soucek, V.

This is more of a complementary answer to Vít Tuček's that addresses the individual questions in the OP. As per his answer, everything that follows below is valid in the more general context of Clifford Algebras in place of $mathbb{H}$.

Functions satisfying $Df=0$ are usually called (left) (quaternionic-)monogenic.

1. 
   Is $ker D$ closed under quaternionic multiplication? - Sadly, no, due to non-commutativity of $mathbb{H}$. Here is an explicit example: define
   $$
   f_k := x_0 e_k - x_k e_0, 1leq kleq 3,
   $$
   where $e_0,dots,e_3$ are the standard basis vectors of $mathbb{H}$. Then $forall 1leq kleq 3: D f_k = 0$, but $forall 1leq kneqellleq 3: D(f_k f_ell) neq 0$.

However, the situation can be somewhat rectified by introducing the so called Cauchy–Kovalevskaya product of two left-monogenic functions. For details about this product see Chapter 14 of Brackx, Delanghe, Sommen - Clifford Analysis (1982).

2. Is this kernel closed under the topology of uniform convergence on compact subsets? - There is a direct analogue of Cauchy's Integral Formula, which eventually leads to establishing that monogenic functions are in particular real-analytic, which in turn leads to the usual Weierstrass Theorems.




3. Does this kernel contain a non-constant bounded function? A kind of Lioville theorem? - Yes, again by real analyticity, an analogue of Liouville's theorem holds: a bounded left-entire monogenic function is constant.




4. Is there a terminology and a classification of all 4-manifolds with an atlas with the property that this operator can be well-defined independently of the choice of a local chart? - Yes, such manifolds are called hypercomplex. I am not aware of any classification of 4-dimensional hypercomplex manifolds (i.e. quaternionic analogues of Riemann surfaces) without further structural assumptions, but then again I am not really familiar with the field, so you should wait for someone else to answer the classification question or post a new question specifically concerning classification issues of hypercomplex manifolds.

It is worth mentioning that despite the negative answer to 1., monogenic functions over hypercomplex manifolds do have an interesting algebraic structure, see for example D. Joyce - Hypercomplex Algebraic Geometry (1998).

5. Indeed, left or right differentiability using the differential quotient approach is satisfied only by linear functions. This and more (including precise references) is explained in Rosenfeld - Differentiable Functions over Associative Algebras (2000).




6. Is the above elliptic differential operator $D$ an appropriate remedy for such technical difficulty? - Yes, studying $D$ and $ker D$ and their generalizations to the Clifford Algebras is the right approach as indicated by Vít Tuček's answer. Some other buzz words include Hypercomplex Analysis and Geometry.