Electrical engineers use them for calculating AC-circuits. In a DC circuit, you only have to worry about how much volt and amperes are in each part of the circuit. In an AC circuit, you also have to worry about the phase, cause the voltage goes up and down. The phase means where in that up and down you are.

The complex number is interpreted as a point on a 2-dimensional plane; the complex plane. You have the "normal" number as 1 axis, and orthogonal to that the imaginary axis. The angle of the vector to that point gives the phase.

They can be generally used for such "wavy" (ie periodical) processes. But I think this particular field of electrical engineering is the main application.

Equations. When we model things with equations, sometimes they don't have a 'solution' at a particular place, unless we use the formal math rules of 'imaginary' numbers like i. Someone else in the comments mentioned electric conductance/resistance in circuits as an example.