First think about what you are really doing with exponents.
If you square a value, you are saying output is from two identical constraints. Like a 45 degree right triangle. The hypotenuse changes based on both sides growing or shrinking the exact same way. That's not to say whatever is doing the squaring is a triangle. It just acts that way. Same with cubing a value. Output is from three identical constraints, like how a surface arc on a sphere changes based on three radii growing or shrinking the exact same way.
Imaginary exponents are just the same. i, 2i, 3i are just like 1, 2 ,3: identity, square, and cube. They just need to run into another imaginary exponent to manifest their value, or you are carrying a lot of extra stuff you don't see.
The reason we like e is because it's derivative is the same as it's function's output. Put in 0, you get 1, and your slope is 1. Now you can start (real,imaginary) at (1,0), and go smoothly to (0,1) without messing with the constants.
This started with Gauss btw. His school teacher asked him to sum the values from 0 to 100, and he saw that as a field of 100, (100+0, 99+1, 98+2 ...)
Some real world examples:
You can harvest an apple orchard by picking the apple from the stem, or cutting the branch. If you cut the branch, you will cut apple blossoms. The apple blossoms are like an imaginary number, and you could make a time based imaginary function that steps out real world apples from the imaginary apples in the blossoms.
Alternating current works by turning off the power in the line intermittently to save power. All they do is switch terminals at the alternator, very fast, 60 times a second, meaning it adds up a lot, with out of phase equipment costing millions more in use. It's like hitting a pendulum on the upswing vs. the downswing. The real value is the 110v you circuit activates at. The imaginary value is whether you are on the upswing and downswing, and if you get a little more, or if you spend money fighting against the phase to close the circuit on the neutral wire.
Stress in odd shape parts with complex loading, like a hip replacement, can flow through with balance, canceling each other out and not stressing the part. Or they can act against each other and build up a lot of internal stress, which is like an imaginary number. Load in the right direction, no matter the magnitude, and all that stress is released, so you're not pressing on metal but a spring releasing away from you dragging you down.