Exactly, you use it to store some information that has no real quantity but may be converted to a real quantity in the future through some other process.
I'm a polytechnic university student, we use imaginary numbers extensively in all sorts of places, especially whenever there is any oscillatory behaviour, such as an electrical signal or a light wave. A complex number is just a two-dimensional vector with real/imaginary components, whcih provides an amplitude and a phase (angle). An oscillating sinusoidal signal/wave may appear to be zero and completely static if you freeze time at the right moment, but as time progresses, it will continue oscillating, like a swing in a park.
In a way, the magnitude represents the built up "momentum" of the system, whilst the real quantity is the immediate physical value at any given point in time (given by the phase). The amplitude is always the same at any given moment, even when the swing is vertical, it has momentum which will help it reach its maximum height.
Personally, I still think they are just "invented", but I think the vast majority of engineers much prefer them to the alternative, manipulating trigonometric functions (every engineer's nightmare). They're a neat way to represent the exchange of potential and mechanical/electrical energy with a single value and some simplified mathematics (this is an engineer's, not a mathematician's, point of view). Like negative numbers, we could have chosen to have two positive quantities, balance and debt, instead we find use in merging these definitions, whether negative values make sense or not. We have become used to to negative numbers representing the "inverse" action, which makes sense when representing a phyiscal quantity such as velocity.