Yes, there is such an analog. Let $U$ be a family of subharmonic functions in a region $D$ subject to the following conditions:
a) For every compact $K\subset D$ there is a constant $C(K)$ such that $u(z)\leq C(K)$ for $z\in K$ and $u\in U$. And
b) For every $u\in U$ we have $u(z_0)>-C$ for some $z_0\in D$ and some real $C$.
Then there is a subsequence which converges to a subharmonic function. The convergence is in $L^1_{\mathrm{loc}}$ and in $D'$ and in many other senses. In particular, Laplacians converge in the weak topology of measures.
If you remove the condition b), one has to add the possibility that the subsequence converges to $-\infty$ (uniformly on compact subsets). Condition b) can be modified in various ways. For example, you can make $z_0$ depending on $u$ but restricted to some fixed compact $K$. All this is true in any dimension.
All this is contained in Theorem 4.1.9 (Hormander, vol. I), or with more details about modes of convergence, in his book Notions of convexity, Birkhauser, Boston, 1994.