Number Lines and Hypercubes

So I am almost finished reading this very interesting book Mathematical Mysteries by Calvin Clawson, and it’s bringing back all the weird math-y stuff I’ve read about over the years and even formally learned a bit of way back in my college days. Primes, e, logarithms, pi, Euler’s Identity, Riemann’s Hypothesis, etc. But one image got me really thinking.

It starts with his explanation of imaginary numbers (the square root of 1, denoted as i). Imagine the real number line (like the kind on your first grade desk, that runs from left to right, zero in the center, positive integers to the right and negative integers to the left). The real numbers are represented by a line.

Now imagine another line, this time perpendicular to the real number line. It’s vertical, so to speak. It represents imaginary numbers, units of i. You sort of now have a two-dimensional graph. All numbers on this graph, this plane, are denoted by two numbers. The first is the real number (where it lies left or right) and the second component is the imaginary part (where it lies above or below the real number line).

If you’ve followed me so far, here is what I started thinking. You got the real number line, a line, going left to right. Then, the imaginary line intersects it, perpendicular to it, going up and down, forming a plane. What type of number would be perpendicular to both, i.e., would form a cube? And what types of numbers would be perpendicular to that, forming a hypercube, a fourth-spatial-dimensional object?

I know I’m not the first person to wonder these thoughts. What I really need to know is, who was, and what was the results of his questioning?

Hmm?