Relation between power of the baseband signal and the low pass equivalent
Since the Hilbert transform is an allpass transformation, the powers of $x(t)$ and its Hilbert transform $\hat{x}(t)$ are the same. Consequently, the complex baseband signal as defined in your question has twice the power of the real-valued bandpass signal. That's why in some textbooks the bandpass signal is defined as
$$x(t)=\sqrt{2}\textrm{Re}\big\{s(t)e^{j\omega_ct}\big\}\tag{1}$$
where $s(t)$ is the complex baseband signal.
The scaling in $(1)$ ensures that $x(t)$ and $s(t)$ have the same power.