Before I studied math, I always slightly resented imaginary numbers as being "math wankery" and just defined because mathematicians had a compulsion to generalize and define new nonsense because they could, and not because it made any sense to.
On the way to changing my mind, I learned that the Fundamental Theorem of Algebra only works for complex numbers (and not "real" numbers), the beauty and simplicity of rotations in the complex plane, but maybe most convincing to me was a history lesson about quaternions.
Quaternions are an extension of the complex numbers, but they're not typically taught in higher math education these days, which contradicted my resentment that mathematicians were just obsessed with getting more and more abstract and general for the sake of it. Of course, they were in vogue in the 19th century (Maxwell's equations were originally written down using them), but mathematicians soon realized they just weren't as useful or as "nice" philosophically as complex numbers and just about anything you can accomplish with quaternions were better accomplished with vectors of complex numbers.
That story played a big part in persuading me that there really is something special about complex numbers -- that maybe they're the most "natural" or "real" numbers of them all.
Also, note the origin story of the complex numbers:
Contrary to legend, they weren't discovered out of a desire by mathematicians to have roots to all quadratics such as x^2 + 1 = 0. It's perfectly sensible for an equation like that to just lack a solution: this just means the standard parabola never drops below zero. Analogously, if we calculate a rocket's payload mass to Low Earth Orbit and the answer is negative, we don't feel a need to find some deep meaning behind negative mass: we just say the rocket can't get to orbit at all. Simple.
It's cubics for which complex numbers were introduced. Cubics (with real coefficients), unlike quadratics, always have real roots, since one arm goes to +∞ and the other to -∞, so it has to cross the x axis somewhere in between. But when the cubic formula was finally discovered, it had this strange property that you frequently had to take square roots of negative numbers, then add those weird square roots to "regular" numbers, and if you just shut up and calculated, the weird parts would always cancel out and you'd get a "regular" number that solved the original equation. That is, you had to pass through the complex numbers in order to find the real solutions.
While Maxwell's equations certainly can be written with quaternions, they were not originally written that way. Maxwell originally just "wrote them out", meaning component-by-component. He had 20 equations! That's why H is sometimes used for magnetic fields: the electric fields were E, F, and G.
Nowadays we usually write 4 vector equations, or 2 in the language of differential forms.
If you do any electronics or signal processing (digital or analog) at all[1], you stop believing that complex numbers are "math wankery" immediately and embrace them as the only thing that's, uh, real.
You mention the Fundamental Theorem of Algebra, I would also add analytic signals. Those are complex by nature, and yet they make so much more sense than real signals ("real" in both senses: non-imaginary and "real world"). In fact, it turns out, real/real world signals are better represented as the sum of two analytic signals.
It's so weird how real numbers, the numbers we consider to be the normal ones, are the special case in a universe that seems to favor complex numbers for the most fundamental things.
[1] Of course also physics in general, but physics can be theoretical, while engineering is almost always rooted in practicality.
The octonions aren’t even associative, making them less “natural” and harder to work with.
But quaternions are quite useful for working with rotations in three dimensions. To be very technical, the unit imaginary quaternions form a double (universal) cover of the rotation group SO(3).
I didn't even know what "Quaternions" were until I started to get into 3D graphics and found that they're often used instead of degrees or radians to avoid camera "gimbal lock".
On the way to changing my mind, I learned that the Fundamental Theorem of Algebra only works for complex numbers (and not "real" numbers), the beauty and simplicity of rotations in the complex plane, but maybe most convincing to me was a history lesson about quaternions.
Quaternions are an extension of the complex numbers, but they're not typically taught in higher math education these days, which contradicted my resentment that mathematicians were just obsessed with getting more and more abstract and general for the sake of it. Of course, they were in vogue in the 19th century (Maxwell's equations were originally written down using them), but mathematicians soon realized they just weren't as useful or as "nice" philosophically as complex numbers and just about anything you can accomplish with quaternions were better accomplished with vectors of complex numbers.
That story played a big part in persuading me that there really is something special about complex numbers -- that maybe they're the most "natural" or "real" numbers of them all.