  This article studies the pointwise convergence for the fractional
Schr\"odinger operator $P^{t}_{a,\gamma}$ with complex time in one spatial
dimension. Through establishing $L^2$-maximal estimates for initial datum in
$H^{s}(\mathbb{R})$, we see that the solution converges to the initial data
almost everywhere with $s>\frac14 a(1-\frac1\gamma)_+$ when $0<a<1$ and
$s>\frac{1}{2}(1-\frac{1}{\gamma})_{+}$ when $a=1$. By constructing
counterexamples, we show that this result is almost sharp up to the endpoint.
These results extends the results of P. Sj\"olin, F. Soria and A. Baily.
Second, we study the Hausdorff dimension of the set of the divergent points, by
showing some $L^1$-maximal estimates with respect to general Borel measure. Our
results reflect the interaction between dispersion effect and dissipation
effect, arising from the fractional Schr\"{o}dinger type operator
$P^{t}_{a,\gamma}$ with the complex time.
