What's the meaning of a complex zero/pole? I have been studying signal processing and control for a while now, and I use Laplace and Fourier transforms almost everyday. Also another tools such as Nyquist or Bode plots.

However, I had never thought of this until today: what is the physical meaning of a complex number when dealing with frequencies?

This may sound silly, but I was asked this question and I didn't know what to answer. Why do we talk about $j\omega$ and not just $\omega$ in, for example, Fourier transforms and Bode or Nyquist plots? What is the physical sense of the real and imaginary part of a zero or a pole in the Laplace domain?
We usually talk of $j\omega$ when we're also interested in the Laplace transform of a signal / system, but want to just talk about the frequency response.

The physical meaning of the imaginary part is that it refers to purely sinusoidal signals and are constant "amplitude". The real part refers to signals for which the "amplitude" decays or grows exponentially.