The identity can be deduced by the Euler integral by homotopy invariance of path integrals.

The function $e^{-z}z^{\alpha}$ is holomorphic on $\mathbb{C}\setminus \{\operatorname{Re}z\le0,\operatorname{Im}z=0 \}$, so the value of the path integral along the boundary of the domain $\{z: \operatorname{Re}z>0,\operatorname{Im}z>0, \epsilon< |z|< \rho \}$ is zero. Hence we get
$$\int_\epsilon^{\rho}i^\alpha e^{-it}t^{\alpha-1}dt=\int_\epsilon^{\rho}e^{-t}t^{\alpha-1}dt+ \int_0^{\pi/2}ie^{-\rho e^{it} }(\rho e^{it})^{\alpha}dt- \int_0^{\pi/2}ie^{-\epsilon e^{it}}(\epsilon e^{it} )^{\alpha}dt $$

We consider separately the three integrals on the RHS.

The first integral, of course, converges to the Euler integral for $\Gamma(\alpha)$ as $\epsilon\to0$ and $\rho\to+ \infty$, provided $\operatorname{Re}\alpha>0$.

The second integrand has absolute value $ e^{-\rho\cos(t) -\operatorname{Im}\alpha t}\; \rho^{ \operatorname{Re}\alpha} $, and it is a bit singular at $t=\pi/2$; to evaluate the corresponding integral, it is convenient to spit it further in the integrals over $[0,x]$ and $[x,\pi/2]$ with a free $x$, optimizing then the bound over the choice of $x$. This way one gets a bound for this integral of order
$O\Big(\rho ^{\operatorname{Re}\alpha -1} \log(\rho) \Big)$, which is $o(1)$ as $\rho\to+ \infty$, provided $\operatorname{Re}\alpha <1$.

The third integrand has absolute value $ e^{-\cos(t) \epsilon-\operatorname{Im}\alpha t}\; \epsilon^{ \operatorname{Re}\alpha}=O(\epsilon^{ \operatorname{Re}\alpha}) $.

We conclude that for $0< \operatorname{Re}\alpha<1$ the integral on the LHS converges, and

$$\int_0^{+\infty} e^{-it}t^{\alpha-1}dt=\frac{\Gamma(\alpha)}{i^\alpha},$$

which is the wanted identity for $z=-1/2$. For any real $z<0$, with a linear change of variable,
$t=(-2z)s$ we plainly get

$$\int_0^{+\infty} e^{2izs}s^{\alpha-1}dt=\frac{\Gamma(\alpha)}{(-2iz)^\alpha}.$$

Finally, for a real positive $z>0$, we take complex conjugate to both sides to the latter identity, written for $-z$ and $\overline \alpha$, which yields to the identity for $z$ and $\alpha$.