And can we have a real frequency-dependent component or constant imaginary component to an impedance?
My thoughts on the first question are that, an imaginary component to the impedance does not dissipate power but it stores it. The imaginary component also introduces a phase shift between the incoming and outcoming voltages. These two facts mean that the power is being stored in some other way, and is then being released again to provide a voltage after some definite time interval. This power is usually stored in a magnetic or electric field (inductor or capacitor respectively)- can we store it in other ways? Perhaps a mechanical flywheel etc- and these systems have a natural resonant frequency which they will oscillate at. By supplying power to these systems, we are in effect driving them, and the power that they will take will be dependent on the driving frequency compared with their own natural resonant frequency. Although I cannot motivate whether all such systems that store power have to have some kind of resonant frequency.
I would guess that this might be the case, because storing energy means having a potential. And, assuming that the undisturbed system was originally in equilibrium, the storing of energy as the system is purturbed away from this point will be in a quadratic potential for small purturbations (ubiquity of the quantum harmonic oscillator).
With regards to having a real, frequency-dependent impedance, I suppose this is possible if a device is, for instance, converting the voltage to motion of some device, and the energy dissipated in this motion is somehow made to be frequency dependent.
In principle I think there should be nothing stopping us having a non-frequency dependent imaginary part to an impedance, but whether this is practically realisable depends on whether we can get a system that stores energy in a non-frequency dependent way. If this energy is being stored in some potential, this would require a constant potential on driving. My intuition says that this would only be possible if we were storing energy in more than one contrasting potetnial, so that the overall potential was approximaely constant. But this would, of course, only work in a limited range.
I can't help but feel that this touches on a more general topic of the meaning of complex numbers in representing physical quantities, which I would be iterested to know more about.