For context (although I will try to phrase this so that it is understandable without the context), I am trying to understand Kronheimer's construction of the ALE (Asymptotically Locally Euclidean) spaces as a hyper-Kahler quotient. Central to this construction, is a quaternion structure on a tensor product of $\mathbb{C}^2$ with the endomorphisms of a complex vector space (which I will define below). I cannot see this quaternion structure for myself, and am quite stuck. I can't find a description of it anywhere and no discussion in this form in any quaternion references I have looked at. In the literature, it seems that this quaternion structure is common knowledge and never explicitly explained. I am also referencing this paper by Anselmi, Billo, and Fre for my working:
https://arxiv.org/pdf/hep-th/9304135.pdf
The setup is to consider $\mathbb{C}^2$ as a representation of $SU(2)$ and the regular representation $\rho_R:\Gamma\rightarrow \text{End}(R)$ of some finite subgroup $\Gamma\in SU(2)$. In my case I'm interested in $\Gamma=\mathbb{Z}_n$.
Anselmi et al claim that the quaternion structure on the space $\mathbb{C}^2\otimes\text{End}(R)$ can realised by writing a pair of endormorphisms $\begin{pmatrix} A\\ B\end{pmatrix}\in \mathbb{C}^2\otimes\text{End}(R)$ as a 'quaternion of matrices' (direct quote):
\begin{equation} p=\begin{pmatrix} A & iB^\dagger\\ iB & A^\dagger \end{pmatrix}, \end{equation} where the dagger refers to the hermitian adjoint. At this point I am thinking of $R$ as a complex vector space where $\dim_\mathbb{C}(R)=n$, so that the endomorphisms $A$ and $B$ can be represented as nxn complex matrices.
When I have tried to check this quaternion structure explicitly, I have rewritten these complex matrices in terms of real matricese as $A=a_1+a_2i$ and $B=b_1+b_2i$ so that $iB=b_1i-b_2$ and $iB^\dagger=-b_1^Ti-b_2^T$ and $A^\dagger=a_1^T-a_2^Ti$. Now I'm assuming these quaternion of matrices have a quaternion structure in the usual sense, that is we define a matrix quaternion $h_p\in M_{n\times n}(\mathbb{H})$ that corresponds to $p$ as $h_p=a_1+a_2i+b_1j+b_2ij=a_1+a_2i+b_1j+b_2k$. We similarly define $\begin{pmatrix} C\\ D\end{pmatrix}\in \mathbb{C}^2\otimes\text{End}(R)$ with a corresponding quaternion of matrices \begin{equation} p'=\begin{pmatrix} C & iD^\dagger\\ iD & C^\dagger \end{pmatrix}, \end{equation} and corresponding matrix quaternion $h_{p'}=c_1+c_2i+d_1j+d_2k$. We can now check $p\times p'$ and compare this against the result of the right multiplication $h_p\times h_{p'}$. Checking the real part we obtain \begin{align} (p\times p)_{11}&=a_1 c_1+i a_2 c_1+i a_1 c_2-a_2 c_2+b_1^Td_1 +i b_1^Td_2 -i b_2^Td_1 +b_2^Td_2 \\ \implies \text{Re}(p\times p)_{11}&=a_1 c_1-a_2c_2+b_1d_1^T+b_2d_2^T,\end{align} which does not match $\text{Re}(h_p\times h_{p'})=a_1 c_1-a_2c_2-b_1d_1-b_2d_2$.
I see two problems: First, I cannot understand why you would take the hermitian adjoint in this instead of simply the complex conjugate like one does for the quaternion structure one puts on $\mathbb{C}^2$ using 2x2 matrices, as even Kronheimer in his original paper goes to great lengths to use this hermitian structure rather than complex conjugation. It may be relevant that later in the construction we are taking only the $\Gamma$-invariant elements of this space, and even later a symplectic quotient of this hyper-Kahler manifold.
Second, why does one need these additional factors of $i$ in the off-diagonal elements which it appears Kronheimer does not use, and these are the source of the sign difference between the terms involving the $b$ and $d$ matrices in this calculation.
Any help at understanding this quaternion structure would be much appreciated (especially the occurrence of the hermitian adjoints instead of simple complex conjugation, the factors of $i$ seem easy to resolve). I am sorry if this question is overly long, I could not figure out how to ask it succinctly.