I had a calculus prof who described it thus:

Calculus is a mathematical model of the world, and it is wrong in the sense of the famous quote about "all models [being] wrong, but some are useful", so mathematicians used the imaginary unit almost like a hotfix for their model.

Whether that's an accurate/rigorous statement or not, it helped me as a learner.

You can, of course, express things like oscillation without i, using vectors and matrices. But the ability to use Euler's formula to express complicated sinusoids as compact exponentials with base e simply makes the analysis much easier. In the words of my prof, it's merely a mathematical structure to express things more easily that "regular calculus" isn't good at expressing.

To me, that made the apparent paradoxes of expressing e.g. phase angles and reactive power and stuff using i a lot easier to stomach.