  In this paper, we study the distribution of the cokernel of a general random
Hermitian matrix over the ring of integers $\mathcal{O}$ of a quadratic
extension $K$ of $\mathbb{Q}_p$. For each positive integer $n$, let $X_n$ be a
random $n \times n$ Hermitian matrix over $\mathcal{O}$ whose upper triangular
entries are independent and their reductions are not too concentrated on
certain values. We show that the distribution of the cokernel of $X_n$ always
converges to the same distribution which does not depend on the choices of
$X_n$ as $n \rightarrow \infty$ and provide an explicit formula for the
limiting distribution. This answers Open Problem 3.16 from the ICM 2022 lecture
note of Wood in the case of the ring of integers of a quadratic extension of
$\mathbb{Q}_p$.
