$G_\mathrm{2D}(\mathbf{x},\mathbf{x_0},\omega) = -\frac{\mathrm{j}}{4}\,H_0^{(2)}(\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|)$
$H_0^{(2)}(\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|)\approx \sqrt{\frac{2}{\pi\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}}\,\mathrm{e}^{-\mathrm{j}\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}\,\mathrm{e}^{+\mathrm{j}\,\frac{\pi}{4}}$ for $ \frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}| \gg 1$
Thus, $$G_\mathrm{2D}(\mathbf{x},\mathbf{x_0},\omega) \approx -\frac{\mathrm{j}}{4}\,\sqrt{\frac{2}{\pi\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}}\,\mathrm{e}^{-\mathrm{j}\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}\,\mathrm{e}^{+\mathrm{j}\,\frac{\pi}{4}}$$ and rearranging $$G_\mathrm{2D}(\mathbf{x},\mathbf{x_0},\omega) \approx \sqrt{\frac{1}{8\,\pi\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}}\,\mathrm{e}^{-\mathrm{j}\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}\,\mathrm{e}^{+\mathrm{j}\,\frac{\pi}{4}}\,\mathrm{e}^{-\mathrm{j}\,\frac{\pi}{2}}$$
$$G_\mathrm{2D}(\mathbf{x},\mathbf{x_0},\omega) \approx \sqrt{\frac{1}{8\,\pi\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}}\,\mathrm{e}^{-\mathrm{j}\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}\,\mathrm{e}^{-\mathrm{j}\,\frac{\pi}{4}}$$If we want a magnitude of 1 at a certain distance $r_\mathrm{ref}=|\mathbf{x_{ref}}-\mathbf{x_0}|$ we have to compensate with $\sqrt{8\,\pi\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}$, which reduces to $\sqrt{8\,\pi\,\frac{\omega}{c}}$ for $r_\mathrm{ref}=1\,\mathrm{m}$.
If we want to compensate for the frequency independent phase shift $-\frac{\pi}{4}$ to obtain the same phase relation as the 3D Green's function we just have to apply a phase correction term $\mathrm{e}^{+\mathrm{j}\,\frac{\pi}{4}}=\mathrm{e}^{-\mathrm{j}\,\frac{7\,\pi}{4}}$.
Note that these correction terms are only valid if $ \frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}| \gg 1$ is fulfilled. In the nearfield of the 2D Green's function a correction term must be obtained from the Hankel function. Nevertheless, we obtain reasonable results if we write
$$G_\mathrm{2D}(\mathbf{x},\mathbf{x_0},\omega) = -\frac{\mathrm{j}}{4}\,H_0^{(2)}(\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|)\,\mathrm{e}^{-\mathrm{j}\,\frac{7\,\pi}{4}}\,\sqrt{8\,\pi\,\frac{\omega}{c}},$$which then yields comparable results of a magnitude 1 @ 1 m corrected 3D Green's function
$$G_\mathrm{3D}(\mathbf{x},\mathbf{x_0},\omega)=\frac{\mathrm{e}^{-\mathrm{j}\,\frac{\omega}{c}\,|\mathbf{x}-\mathbf{x_0}|}}{4\,\pi\,|\mathbf{x}-\mathbf{x_0}|}\cdot 4\,\pi\,.$$