Complex Morlet coefficient
Your $C$ is here for the wavelet energy normalization: it is chosen such that:

$$ \int \psi(t)^2dt = 1\,,$$

the most  common framework for wavelets. It ensures that wavelet coefficients can more or less directly be interpreted in energy. The problem is that, as is, the wavelet is not admissible. If you take $\omega \simeq 0$, it reduces to a Gaussian, and does not  oscillate enough to have zero-mean.

However, when $\omega$ becomes "big enough", some  say $\omega&gt;5$ (a common choice is $\omega = \pi\sqrt{2/\log2} $), one usually considers that it is "numerically admissible" (with high precision). 

A more proper version (with $\omega_0$ instead of $\omega$ to avoid strange interpretations) is: 

$$ \psi_{\omega_0} = \frac{C_{\omega_0}}{\pi^{1/4}} e^{-t^2/2}(e^{i\omega_0 t} -e^{-\omega_0^2/2})$$

and then you get:

$$ 1/C_{\omega_0}^2 = 1+e^{-\omega_0^2} -2e^{-3 \omega_0^2/4} \,,$$

which is very close to $1$ when $\omega_0$ is big enough.