It is well known that $SU(2)$ and the unit quaternions $S^3\subset\mathbb{H}$ are isomorphic Lie groups.
SU(2) has a standard representation on $\mathbb{C}^2$ (via left matrix multiplication) and $S^3$ has a standard representation on $\mathbb{H}$ via left multiplication.
Are these representations isomorphic? (here they just affirm that it is the quaternionic multiplication without specifying if they mean the left or the right one https://en.wikipedia.org/wiki/Representation_theory_of_SU(2)#Most_important_irreducible_representations_and_their_applications)
What is the explicit isomorphism of $\mathbb{C^2}$ and $\mathbb{H}$ which realizes such isomorphism?