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.

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

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)*.

**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.

**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.

**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)*.

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)*.

**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.