The Morse complex $\mathcal{M}(Δ)$ of a finite simplicial complex $Δ$ is the complex of all gradient vector fields on $Δ$. In this paper we study higher connectivity properties of $\mathcal{M}(Δ)$. For example, we prove that $\mathcal{M}(Δ)$ gets arbitrarily highly connected as the maximum degree of a vertex of $Δ$ goes to $\infty$, and for $Δ$ a graph additionally as the number of edges goes to $\infty$. We also classify precisely when $\mathcal{M}(Δ)$ is connected or simply connected. Our main tool is Bestvina-Brady Morse theory, applied to a "generalized Morse complex."