As others have pointed out, there are definitely real world applications for i, so I won’t bother to reiterate them, but I also don’t think that that’s the issue here. Math, in order to make intuitive sense, has to be able to have some kind of line connecting a new concept back to ones that already make intuitive sense. The unfortunate part about imaginary numbers is the choice of the name, which implies that they don’t have real world applications. Worse, while it is helpful to have real world applications for mathematical concepts, most of the i is useful for aren’t something that is really appropriate for when the mathematical concept. So, instead, we get stuck saying “trust us, it’s useful.”
The natural explanation, comparing complex numbers to negative numbers, I think ultimately just confuses the issue. I have 5 ducks, that’s a positive number, then I have 3, so that difference is conceptually equal to -2 ducks, that’s a negative number. But even knowing that i is the square root of -1, that doesn’t give us a useful conceptual analogy, because subtraction isn’t a place where that concept would be used.
Instead, I think a better analogy would be considering space. We can all imagine one dimension as a number line, two as a cartesian plane, and three dimensions as x, y, z. We intuitively can carry on that concept to four or more dimensions and come up with useful applications. For instance, adding t gives you space-time, or we could add a w and get a fourth spatial dimension, or whatever. I think it’s fairly easy to see how we could conceptually expand that to any arbitrary number of dimensions for various applications, but those are a couple simple ones. However, despite knowing that there’s intuitive real world applications for hyperspace, we can’t meaningfully model even very simple concepts. A hypercube is virtually impossible to wrap one’s head around.
So this is how I think about complex numbers. Take a coordinate like (3,4); I intuitively know what that is and how to find it on graph. But I could also sort of think of the axes such that “x” is a unit of left-right-ness and “y” is a unit of up-down-ness. Thus, with just a bit of a difference in notation, I could think of it as 3x + 4y. So, if we can have an arbitrary number of extradimensions, why can’t one of them be i instead of z or t or whatever? In just the same way that a particular dimension is useful in certain applications, like t is useful when time is relevant, and ignored when it’s not, or z is useful when dealing with three spatial dimensions and ignored when it’s not, i is useful in applications where the square root of negative numbers comes into play. Thus, taking the same concept above, you get something like 5 + 6i.
Beyond that, the difficult part is that complex numbers don’t follow the same sorts of rules that your rationals do. But really, it’s the whole point that the rules they follow work differently that makes them a bit different. It’s been a few years since I worked on it, but I used them extensively in my work on my PhD thesis because I was working on modeling human movement, so they showed up a lot when working with angular velocity and such. I imagine that, considering both are related to the right-hand rule, that’s probably how it comes up in electromagnetism too, but it’s been far longer since I did any of that.