Holomorphic Map of Quaternion Algebra

$\newcommand{\gi}[1]{\boldsymbol{#1}}$

Let $E$ be identity map of quaternion algebra and $I$, $J$, $K$ be maps of conjugacy of quaternion algebra. The derivative of map
$$f:H\rightarrow H$$
of quaternion algebra has form
$$\begin{split} \frac{\partial f}{\partial x}=&-\frac 12 \left( \frac{\partial f}{\partial x^{\gi 0}} +\frac{\partial f}{\partial x^{\gi 1}}i +\frac{\partial f}{\partial x^{\gi 2}}j +\frac{\partial f}{\partial x^{\gi 3}}k \right)\circ E \\& +\frac 12\left( \frac{\partial f}{\partial x^{\gi 0}} +\frac{\partial f}{\partial x^{\gi 1}}i \right)\circ I +\frac 12\left( \frac{\partial f}{\partial x^{\gi 0}} +\frac{\partial f}{\partial x^{\gi 2}}j \right)\circ J \\& +\frac 12\left( \frac{\partial f}{\partial x^{\gi 0}} +\frac{\partial f}{\partial x^{\gi 3}}k \right)\circ K \end{split}$$

The map of quaternion algebra which satisfies the equation
$$\frac{\partial f}{\partial x^{\gi 0}}+i\frac{\partial f}{\partial x^{\gi 1}} +j\frac{\partial f}{\partial x^{\gi 2}}+k\frac{\partial f}{\partial x^{\gi 3}}=0$$
is called left-holomorphic. The map of quaternion algebra which satisfies the equation
$$\frac{\partial f}{\partial x^{\gi 0}}+\frac{\partial f}{\partial x^{\gi 1}}i +\frac{\partial f}{\partial x^{\gi 2}}j+\frac{\partial f}{\partial x^{\gi 3}}k=0$$
is called right-holomorphic.

Evidently that maps $I$, $J$, $K$ are left-holomorphic and right-holomorphic. However, there exist not trivial examples of holomorphic map.

Let $\mathcal HE$ be set of maps which satisfy the equation
$$\frac{\partial f}{\partial x^{\gi 0}} +\frac{\partial f}{\partial x^{\gi 1}}i = \frac{\partial f}{\partial x^{\gi 0}} +\frac{\partial f}{\partial x^{\gi 2}}j = \frac{\partial f}{\partial x^{\gi 0}} +\frac{\partial f}{\partial x^{\gi 3}}k =0$$
Let $f$, $g\in\mathcal HE$. Then $f\circ g\in\mathcal HE$. Maps which belong to the set $\mathcal HE$ should have interesting properties and it is mind to study this set. Holomorphic map $f\in\mathcal HE$ satisfies the equation
$$\frac{\partial f^0}{\partial x^0}=0$$