Imagine getting to a solution is a pathway and you walk along it to find a solution, you find a tree in the path there is no way to get to the other side you could give up and say it is not possible to get to the solution but that is not true you could leave the path and walk around or climb over.

Imaginary numbers give you a way to leave the real numbers and find a solution. The solution is not always imaginary just because you used them to get it.

Imaginary is a bad name because they are actually numbers and the real numbers are just a subset of them.

We use imaginary numbers a lot in engineering, it would be impossible for modern control systems and power networks to work without them.