We identify $\mathbb{R}^4$ with the quaternions $\mathbb{H}=\{t=x+yi+zj+wk\mid x,y,z,w\in \mathbb{R}\}$. We define the differential operator $D$ on $C^{\infty}(\mathbb{R}^4)$, the space of smooth quaternion-valued maps, via
$$
D(f):= \frac{\partial}{\partial\bar{t}}f,
$$
where notice that $\partial/\partial\bar{t} = \partial/\partial x+ i\partial /\partial y +j\partial/\partial z+k\partial /\partial w$.



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> 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?)
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> 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?
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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 $\lim\_{t\to t\_0} \frac{f(t)-f(t\_0)}{t-t\_0}$.



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> Is the above elliptic differential operator $D$ an appropriate remedy for such technical difficulty?
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Quaternion holomorphic maps via certain elliptic operator instead of immediate generalization of complex differentiability
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.


