For $$\gamma >0$$ and $$a>0,$$ the operator $$P_{a,\gamma }^{t}f$$ of Schrödinger type with complex time is defined by $$\begin{aligned} P_{a,\gamma }^{t}f(x)=S_{a}^{t+it^{\gamma }}f(x) =\int _{{\mathbb {R}}} e^{ix\xi }e^{it|\xi |^{a}}e^{-t^{\gamma }|\xi |^{a}} {\hat{f}}(\xi )d\xi , \end{aligned}$$and the corresponding maximal operator $$P_{a,\gamma }^{*}$$ is defined by $$\begin{aligned} P_{a,\gamma }^{*}f(x) =\displaystyle \sup _{0<t<1}|P_{a,\gamma }^{t}f(x)|,\quad x\in {\mathbb {R}}. \end{aligned}$$When $$0<a<1$$ and $$\gamma >1,$$ some characterization of the global $$L^{2}$$ estimate for the maximal operator $$P_{a,\gamma }^{*}$$ is obtained. The authors extend the results of the maximal operator $$P_{a,\gamma }^{*}$$ for $$a>1$$ and $$\gamma >1$$ in Bailey (Rev. Mat. Iberoam 29: 531-546, 2013).