In the category of symplectic algebraic varieties moment maps have in general disconnected fibers. Easy example go as follows: Let $T={\bf G}_m$ act on the affine plane ${\bf A}^2$ by $t\cdot(x,y)=(tx,t^{-1}y)$. Then symplectic form $\omega_0=dx\wedge dy$ is $T$-invariant. The corresponding moment map is $m_0(x,y)=xy$.
Let $X_1:={\bf A}^2\setminus\{(0,0)\}$. Then the zero-fiber of $m_0$ becomes disconnected.
A less trivial example is as follows: Let $f(z)$ be an arbitrary non-constant polynomial with derivative $f'(z)$. Now rescale the symplectic form to $\omega=f'(xy)\omega_0$. Then $\omega$ is nondegenerate on the open subset $X_2=\{f'(xy)\ne0\}\subseteq{\bf A}^2$. The corresponding moment map is $m(x,y)=f(xy)$. So, unless $f$ is linear, the generic fibers of $m$ are disconnected.
The second example can be generalized as follows: Start with any moment map $m:X\to\mathfrak t^*$. It is dominant if the action is effective. Let $Y\to\mathfrak t^*$ be any é