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IIRC from my physics class in college a long time ago, this isn't even a "somehow." It's that complex numbers were formulated and used because they were useful in physics, specifically for modeling behaviors of real-world objects, not some obscure electromagnetism effects happening in a circuit or whatever. I wish I could remember the details and/or how apocryphal the story was, but it's certainly one of the less intuitive things that square-roots of negative numbers are so useful in real-world physics, looking at math from the outside.
This isn't true. They were originally developed to solve equations. The physics applications came much later
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Nah; the whole reason imaginary numbers are called "imaginary" is that they were first used in formal, temporary, intermediate results in algorithms for calculating the "real" cubic/quartic polynomial roots that people care about. That was like 1600. I think Euler's formula a century later was when engineers and physicists first really started treating complex numbers as things of non-temporary interest, and quantum mechanics was when complex numbers started to feel more "real" than real ones.
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