The complex numbers are solution for mathematics equation.
The physics meaning of complex number depends on how and where you use it.
For example: There is a 90 degree phase between voltage and current for inductor and capacitor.
The inpedance for an inductor is represented as j\omega L, and impedance for a capacitor is represent as \frac{1}{j\omega C}, where j=\sqrt{-1}.
For an R-L-C series circuit, the total impedance is Z=R+j(\omega L-\frac{1}{\omega C}).
The impedance Z is a complex number which means that there is a phase shift between current flow thrugh the circuit and the total voltage applied to the circuit.

Many calculation become easier with the help of complex number.

We need to solve differential equation for many different problems.
Some of those might not be easy to solve.
However, if we make a Laplace/Fourie transform, the differential/integral equation become multiple/divide operation.
In many cases, it is much easier to solve it this way and find out it's solution.
Then, we can make a inverse transformation to get it solution to the coordinate system we are more familiar with.

For example: We measure sound wave as a function of time and the singal can be shown in oscilloscope.
However, it is not easy to find information from the sound wave directly.
If the sound wave was being transofrmed to frequence space (Fourier transform). Then, it is much easier to find out the characteristic pattern for different music instrument ( and different personal have unique frequence pattern,too). So there are many different transformation tools help us to transoform our data from one representation to another representation (Just like look at the same event from different point of view).
It seems that engineer use more Laplace transform(decay system) and physicist use more Fourier transform (oscillation system).