Let \(f \in \mathcal O(\mathbb C^n)\), \(n \geq 2\), be a non-constant holomorphic function with \(df\) not identically zero on the level set \(X_0 = f^{-1}(0)\) and such that \(\tilde H^{n-2}(X_0) = 0\). Let also \(\overline{\mathbb C^n_f}\) be the suspension of \(\mathbb C^n\) by \(f\), i.e., the hypersurface of \(\mathbb C^{n+2}\) defined by the equation \(z_{n+1} z_{n+2} = f(z_1, \ldots, z_n)\). In this paper, the author first presents a new criterion for checking whether a complex manifold satisfies the volume density property (shortly, VDP), i.e., whether the Lie algebra of complete divergence free vector fields is dense in the full Lie algebra of divergence free vector fields. Secondly, he uses such new criterion to show that each of the above described suspensions \(\overline{\mathbb C^n_f}\) has the volume density property with respect to the volume form \(\widetilde \omega\) determined by the condition \(d(z_{n+1} z_{n+2} - f) \wedge \widetilde \omega = d z_1 \wedge \ldots \wedge d z_n \wedge d z_{n+1} \wedge d z_{n+2}\). By appropriately choosing \(f\), this result gives the first examples of manifolds with VDP that are not biholomorphic to any algebraic manifold. The new construction gives also potential counterexamples to the Zariski Cancellation Problem.