I was proud of this idea, as it was probably the best early research idea I've come up with. I became very interested in matrices and linear algebra after reading a paper on modelling origami using rotation and translation matrices (Belcastro and Hull, 2012). I began to play with expressing all kinds of things as matrices and seeing what "meanings" matrix operations had in those contexts. Somehow complex numbers cropped up, and I decided that they were a good candidate for this "interpretation" because multiplying by a complex number meant a rotation and dilation of the complex plane  or an "amplitwist" (Needham, 1996). So I represented a complex number as a rotation matrix together with a scaling factor. Given a complex number $z = r\mathrm{e}^{\mathrm{i}\theta}$, let $\mathbf{Z} = r\begin{bmatrix}\cos\theta&\sin\theta\\\sin\theta&\cos\theta\end{bmatrix}.$ Some interesting properties follow immediately from this correspondence:
I liked to write (5) as $\lvert\mathbf{Z}\rvert = \lvert z\rvert^2$, using the "absolute value" notation for matrix determinants. In terms of Abstract Algebra, the next two properties are more important. Suppose we have complex numbers $a$ and $b$ that correspond to matrices $\mathbf{A}$ and $\mathbf{B}$ respectively.
The multiplication property comes directly from how I defined the "matrix correspondence": by treating matrix multiplication as a matrix transformation. The addition property was unexpected, but on hindsight it came from the fact that a complex number $x + y\mathrm{i}$ would correspond to $\left[\begin{smallmatrix}x & y \\ y & x\end{smallmatrix}\right].$ Prove all of the properties mentioned above. How would you generalize this "expansion" of a complex number further? I have some hints if you're stuck...
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