When do you need a complex termination?
So the characteristic impedance is a complex number. But why is never
a complex termination used?

When \$j\omega\$ is significant, the formula reduces to this: -
$$Z_0 = \sqrt{\dfrac{j\omega L}{j\omega C}} = \sqrt{\dfrac{L}{C}} = \text{resistance}$$
This applies pretty much to frequencies of 1 MHz and greater i.e. the vast majority of RF applications.
At audio frequencies, the equation becomes more complex and is usually approximated to this: -
$$Z_0 = \sqrt{\dfrac{R}{j\omega C}}$$
So, for telephony applications, a complex termination method is used (in order to reduce telephone side-tone) like this: -

New Zealand and the UK use the above complex impedance for terminating a telephone cable in order to optimize the anti-side-tone circuit. Other countries use similar but not identical values.
Generally, a cable will exhibit this type of impedance response: -

At DC (well, 30 Hz) the impedance is determined by the ratio \$\sqrt{\frac{R}{G}}\$ and if R = 1 Ω then, according to the above (taken from wiki), \$\frac{1}{G}\$ has to be 1.6 GΩ and this wouldn't surprise anyone I would think! But, it all gets a little messy at mid-band audio and up to about 100 kHz for most cables.