Such extension exists. This follows from the followoing three facts.

1) For every point $x\in X$ there is a neighbourhood $U(x)\subset M$ and a holomorphic map
$\phi: U(x)\to \mathbb C^n$ such that $\phi (U(x)\cap X)\subset \mathbb R^n$.

2) Let $\alpha$ be a real analytic two-form defined on $\mathbb R^n$, then it can be extended to a holomorphic two-form on a neighbourhood of $\mathbb R^n$ in $\mathbb C^n$ that retracts on $\mathbb R^n$

3) If you have two extensions of $\alpha$ to a same neighbourhoods of $\mathbb R^n$ then they coincide.

I don't know a reference for 1) but it should be something standard, to get 2) and 3) you can just use Taylor decomposition of $\alpha$.

Once you know 1,2,3 the statement follows easily.