The article
Alesker, S. (2003). Quaternionic Monge-Ampere equations. The Journal of Geometric Analysis, 13(2), 205-238.
has the following CLAIM:
Claim. Let $A$ be an invertible hyperhermitian matrix of order $n$. For any $i$, $1\le i\le n$, $$\left(A^{-1}\right)_{ii} = \frac{1}{\det A} \det M_{ii}(A) .$$
$M_{ii}$ denotes the minor of a quaternionic matrix $A$ obtained from $A$ by deleting the $i$-th row and the $i$-th column.
I also found how to get the inverse matrix in:
Chen, L. (1991). Inverse matrix and properties of double determinant over quaternion field. Sci. China Ser. A, 34(5), 528-540.
But I still don't know how to show the claim. Any help?