Up until the end I was waiting for the revelation that what we see is really just the rotation around the complex unit circle, and just like there we put in the i as the factor in the exponent, the scaled exponent [0, 1; -1, 0] you chose is also the thing that by multiplication creates a counterclockwise right angle turn, and just like e^it, the value is the factor that does a rotation about any chosen angle.
TOM ROCKS MATHS 
Up until the end I was waiting for the revelation that what we see is really just the rotation around the complex unit circle, and just like there we put in the i as the factor in the exponent, the scaled exponent [0, 1; -1, 0] you chose is also the thing that by multiplication creates a counterclockwise right angle turn, and just like e^it, the value is the factor that does a rotation about any chosen angle.
Maybe in a follow-up?
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