The axes are drawn sideways

Flip the axes Wessel's way — and $e^{i\theta}$ starts rotating clockwise, like a compass, a ship's wheel, and a vessel's heading.

Full paragraph (abstract translation)

The complex plane was first given geometric form by Caspar Wessel (1799), and his background explains a lot: Wessel was a Norwegian surveyor, and his motivation was an analytic representation of direction. In a surveyor's world the natural orientation is navigational: forward is up, "to the side" is right, and positive rotation follows the hands of the clock.

Argand's diagram, however — the one that made it into the textbooks — chose a Cartesian orientation: the real axis horizontal, positive rotation going counter-clockwise.

Restore Wessel's system: point the real axis upward and the imaginary axis to the right. Positive rotation from $\mathrm{Re}$ to $\mathrm{Im}$, algebraically the same, becomes clockwise rotation. The complex exponential $e^{i\theta}$ now turns together with the clock hand, with the compass going from north to east, and with a ship's heading bearing to starboard. Gauss's direct and lateral are recovered as forward and sideways.

The angular debt inherited from Cartesian diagrams — not from the geometry itself — simply evaporates.

Why it matters

• In navigation, robotics, and motor control, "angle" is almost always clockwise from north. Mathematical physics, by habit, insists on translating everything into "counter-clockwise from east." Every such translation is a source of sign errors.
• Wessel's convention makes $\cos\theta + i\sin\theta$ the very same motion as a compass's $\mathrm{heading}(t)$, with no corrections.
• The algebra of $\mathbb{C}$ does not change: only the picture on the blackboard does.

Reference

• Wessel, C. (1799) Om directionens analytiske betegning. Nye samling af det Kongelige Danske Videnskabernes Selskabs Skrifter, 5, 469–518. sophiararebooks.com