Erg. It's true that complex numbers model a rotation operator in 2d geometry. But I think it's harmful to say that it's better to think about 2d geometry in terms of complex numbers rather than, well, 2d geometry (vectors and rotation operators/matrices/geometric algebras). This kind of thinking doesn't extend to higher dimensions, obfuscates the actual discovery under the 'magic' of complex numbers, and requires knowing 'one weird trick' to figure out yourself.

(edit: added reference to geometric algebras, which are cleaner than rotation matrices for intuitively modeling this.)

I still don't get the significance of complex numbers, to me they seem like vectors with certain operations and I still suppose there is no deeper revelation, that sqrt(-1) represents, and every example could have a geometric interpretation, even if this isn't motivated by the example. Likewise, the only explanation for e^(pi*i)-1=0 I know is the conversion to polar coordinates.

>Since you need to keep track of your current orientation this slightly complicates things a bit

>If at each step we keep the imaginary multiplication term from the previous step we can effectively encode the rotation in our expression

Why, you could just keep track of the rotation as an angle, which would likewise be a result of the previous step. In binary there wouldn't even be much of a difference: Look at a 2-bit value either as a modulo 4 ring of type pi/2, or as a modulo 2 vector of the two complex components. That's my first thought, at least. Although, the real and imaginary parts would always have opposite values, so only one bit is needed. I wonder if complexity analysis would show a difference.

Good stuff! Even though this may not be the best application for complex numbers, it is a very simple example for a practical use. For some person out there, this may even serve as a "gateway drug" to complex analysis.

Here is the pseudo-code for matrix rotation for comparison:

  def rotate(vector, angle):
    rotation_matrix = [
      [cos(angle), -sin(angle)],
      [sin(angle), cos(angle)]
    ]
    return rotation_matrix * vector
  
  position = (0, 0)
  velocity = (0, 1)
  for instruction in [x.strip() for x in input.split(",")]:
    direction = instruction[0]
    distance = int(instruction[1:])
    angle = 90 if direction == 'L' else -90
    velocity = rotate(velocity, angle)
    position += velocity * distance
  print position