  The purpose of this paper is to study the unique continuation property for a
Schr\"odinger-type equation $ \bar\partial u = Vu$ on a domain in $\mathbb
C^n$, where the solution $u$ may be a scalar function, or a vector-valued
function. While simple examples show that the unique continuation property
fails in general if the potential $V\in L^{p}, p<2n$, we first prove that, in
the case when $u $ is a scalar function, the unique continuation property holds
when $V\in L_{loc}^{2n}$ and is $\bar\partial$-closed. For vector-valued smooth
solutions, we establish the unique continuation property either when $V\in
L_{loc}^p $, $ p>2n$ for $n\ge 3$, or when $V\in L_{loc}^{2n}$ for $n = 2$.
Finally, we discuss the unique continuation property for some special cases
where $V\notin L_{loc}^{2n}$, for instance, $V $ is a constant multiple of $
\frac{1}{|z|}$.
