On a class of automorphisms in \mathbb{H}^2 with volume

We briefly present the theory of slice-regular functions of one quaternionic variable and compare this regularity properties with Fueter regular functions (the regularity used physics).

We give a possible extension for shears and overshears in the case of two non-commutative (quaternionic) variables in relation with the associated vector fields and flows. We present a possible definition of volume preserving automorphisms, even though there is no quaternionic volume form on \mathbb{H}^2.

Using this, we determine a class of quaternionic automorphisms for which  the Andersen-Lempert theory applies. Finally, we exhibit an example of a quaternionic  automorphism, which is not in the closure of the set of finite compositions of volume preserving quaternionic shears, though its restriction to complex subspace is in the closure of the set of finite compositions of volume preserving complex shears.