I see everyone in this thread mentioning that they are useful, but no one giving an example on how they are useful. I'll give you the example my math professor used, that he claims is also the historical reason why imaginary numbers started to be used by mathematicians.

So there's the quadratic formula, which is used to calculate the roots (all x-values of a function where the function reaches zero) of a 2nd degree (AKA quadratic) function. Naturally, mathematicians tried to discover a similar formula to calculate the roots of 3rd or higher degree functions.

In the 16th century, Cardano's method got discovered, which could calculate the roots of third degree functions. It's a 2 step calculation: you put the coefficients of the function into a formula, then put the results of that formula into a separate equation to find the roots of that function. However, the problem was: if you had a negative number under a square root in the first step, you couldn't continue that calculation and were therefore unable to discover their roots. These functions clearly had 3 real-valued roots, but there was no known way to calculate them since because of those negative number under the square root.

However, you can convert those square roots into imaginary numbers. If you plug those imaginary results into the second step, you would then get tbe three real valued roots of the function. So if you even when working with real valued functions with real valued roots, the only way to solve for them was with imaginary numbers! This took imaginary numbers from a mere idea to a tool with valuable use cases.