If you do want to think of geometry in terms of linear algebra structures, you probably want to be studying Geometric Algebra.
My point is, I think it's harmful to say "hey, you - you're trying to solve a problem in 2d space? Have you considered using complex numbers?" When the correct thing to do is "have you considered using the algebra of N-d spaces?" (that is, Clifford/geometric algebra), and leave all the magical complex numbers out of it.
It's easy to reach for complex numbers when they're built-in. Do you have to use Maple/Mathematica for this stuff, or can you do it easily in Python/JS/&c ?
You can find the first chapter as a pdf download here: http://geometricalgebra.org/downloads/ga4cs_chapter1.pdf. It contains some code examples, and the complete source code is here: http://geometricalgebra.org/sandbox.html
The GACS book is indeed great. The “conformal model” defined there is very pleasant to work with. If you want to learn about it without buying a textbook, there’s also this PhD thesis: http://www2.eng.cam.ac.uk/~rjw57/pdf/r_wareham_pdh_thesis.pd...
Even more lightweight than matrices is just implementing the transformation on vector objects yourself: a 90-degree rotation left is R(x,y) = (-y, x), and (-R)(x,y) = -(R(x,y)). In higher dimensions this remains true for any 2-dimension subplane, so R_zx(x,y,z) = (z,y,-x).
Rotation matrices / other formulations become necessary when you're dealing with being able to rotate through any angle.
Though, doing rotations of the matrix 'y = Rx' form suffers mathematical problems like numerical instability and bad behavior. Quaternion rotations or Geometric Algebra's rotors (which are equivalent for N=3, and both take the form y = RxR^-1) work a lot better, and also eliminate a lot of the redundancy in the (skew-symmetric) rotation matrices formulation (4 variables instead of 9, for 3d).
What I mean is that the intuition around complex numbers doesn't extend. Certainly not the "i is the square root of -1"-level understanding of complex numbers.
Clifford algebras are better thought of as structures built over the higher-dimension vectors spaces, with little relationship to algebras/fields. They have much more in common with the matrix representation than the algebraic one. The algebraic perspective only supports complex, quaternion, and octonion numbers before breaking down.
(edit: added reference to geometric algebras, which are cleaner than rotation matrices for intuitively modeling this.)