  In this paper, first we consider the uniform complex time heat kernel
estimates of $e^{-z(-\Delta)^{\frac{\alpha}{2}}}$ for $\alpha>0, z\in
\mathbb{C}^+$. When $\frac{\alpha}{2}$ is not an integer, generally the heat
kernel doest not have the Gaussian upper bounds for real time. Thus the
Phragm\'en-Lindel\"of methods fail to give the uniform complex time estimates.
Instead, our first result gives the asymptotic estimates for $P(z, x)$ as $z$
tending to the imaginary axis. Then we prove the uniform complex time heat
kernel estimates. Finally we also show the uniform estimates of analytic
semigroup generated by $H=(-\Delta)^{\frac{\alpha}{2}}+V$ where $V$ belongs to
higher order Kato class.
