We define the atomic Hardy space $H^p_{\mathcal{L},\operatorname{at}}(\mathbb{C}^n)$, $0\lt p≤ 1$, for the twisted Laplacian $\mathcal{L}$ and prove its equivalence with the Hardy space defined using the maximal function corresponding to the heat semigroup $e^{-\mathcal{L}t}$, $t\gt 0$. We also prove sharp $L^p$, $0\lt p≤ 1$, estimates for $\mathcal{L}^{β/2}e^{i\sqrt{\mathcal{L}}}$. More precisely, we prove that it is a bounded operator on $H^p_{\mathcal{L},\operatorname{at}}(\mathbb{C}^n)$ when $β≥(2n-1)(1/p-1/2)$.