In my opinion, complex numbers are between scalars and vectors. (There are others there too, like quaternions.)
The \$i^2 = -1\$ thing is just a "mathematical" way of expressing the idea of rotation by 90°. If you think of the one-dimensional number line concept, then such a rotation by itself doesn't make any sense, but if applied twice, you mirror the number line. And that's exactly what happens.
The math boffins examined the concept for a bit, and realized that it also naturally aligned with the exponential function, so that \$e^{a + i b}\$ corresponds to a scaling (multiplication) by \$e^a\$, and rotation by \$e^{i b}\$. Everything else stems from there.
So, while the "imaginary number" concept doesn't make much sense for real numbers (integers, rationals, irrationals), it does work as a stepping stone for extensions to more complex concepts.
In certain ways, complex numbers remind me of homogenous coordinates. Cartesian coordinates \$(x_1, \dots, x_N)\$ are represented by \$(X x_1, \dots, X x_N, X)\$ in homogenous coordinates, and somewhat similarly to complex numbers, homogenous coordinates can be used to express both translation and rotation (full transform) in a single operation. (In fact, the vast majority of computer 3D graphics libraries use homogenous coordinates at least internally for transformations.) A particularly interesting facet of homogenous coordinates is that they let one trivially distinguish between
position vectors (\$(x_1, \dots, x_N, 1)\$) and
direction vectors (\$(x_1, \dots, x_N, 0)\$). Mathematically, there is no reason to distinguish between the two, but in 3D visualization, the ability to easily do so while applying the exact same transformations (with translation only affecting position vectors and not direction vectors) simplifies things quite a bit.
So what the hell are the irrationals, then?
Any number that you cannot express as the ratio of two integers (or an extension of the same concept for e.g. complex numbers).
A step further are transcendental numbers. They are numbers you cannot obtain as a root of a finite-degree nonzero polynomial with rational coefficients, like \$\pi\$ and \$e\$.
But you have heard about those "atoms" and "quantum" stuff, right?
There is nothing real about real numbers.
The two 'real' mean completely different things: one is 'physically real', and the other is 'the set of numbers \$\mathbb{R}\$ that we call "reals"'.
The theory that they are an inside joke of heterosexual European men invented to confuse Americans doesn't even seem so far off 
Except they weren't initially European and never even "mostly" European, men, or even heterosexual.
That's what makes "decolonizing math" so ridiculous, and extremely racist and bigoted concept.
In particular, Indians and Arabs are quite,
quite offended by your suggestion, and for good reason. Sthananga Sutra is about 2400 years old, and covers things like fractions and elementary number theory. In the 800s Arabs made huge leaps forward in maths, especially people like Muhammad ibn Musa al-Khwarizmi.