Imaginary numbers were useful as soon as they were invented. They are algebraically closed, unlike the real numbers, and are required for the fundamental theorem of algebra.

https://en.m.wikipedia.org/wiki/Fundamental_theorem_of_algeb...

The first use of negative roots was to find the (real) roots of certain polynomials. They were considered a mathemathical hack: useful, but specious on their own. And then it turned out that if you take them on their own terms, they're fantastically useful.

In addition to the other points, imaginary numbers immediately added to the representation of things we could represent. Simply rewriting existing numbers in other ways isn't all that interesting; it doesn't add anything. We already knew there's an infinite number of ways to serialize numbers to symbols. Negative base numbers don't behave differently than positive base numbers, because "negative base numbers" and "positive base numbers" are artifacts of English and don't actually exist mathematically. There are numbers, which are written in different representations, but are not actually different entities.

So there's no possible way this could be useful for "higher dimensional math" or anything, because there's no entity being added here. Higher dimensional math already uses numbers. At most there might be one or two concepts that might in some manner if you squint hard enough could be better represented by negative base numbers, but even then the positive base representation would probably still be how people actually understood it.

> At most there might be one or two concepts that might in some manner if you squint hard enough could be better represented by negative base numbers,

That's what I was thinking.

It will be really funny if there is some bit of new, undiscovered physics, that went undiscovered for so long just because it's best described using some esoteric maths like complex base numbers.

Highly unlikely, but fun to think about.

Are there any examples where the representation of a number matters in physics?

I recall some effort in moving from imperial measurement to metric. Reality itself does not care, but the math sure gets easier if you carefully select units.

Actually, another neat example is analog vs digital computation. your models can get very different answers. I recall some Mandelbrot guy talking about that.

See Gaussian (cgs) units for an example, compare Coulomb's law in cgs and SI: https://en.wikipedia.org/wiki/Gaussian_units#Unit_of_charge