  In this paper we first prove an isomorphism between certain spaces of Jacobi
forms. Using this isomorphism, we study the mod $p$ theory of Hermitian Jacobi
forms over $\mathbb{Q}(i)$. We then apply the mod $p$ theory of Hermitian
Jacobi forms to characterize $U(p)$ congruences and to study Ramanujan-type
congruences for Hermitian Jacobi forms and Hermitian modular forms of degree
$2$ over $\mathbb{Q}(i)$.
