Personally I take “similarity transformations which fix a point in the plane” (or if you like, “amplitwists of planar vectors”) to be the definition of complex numbers, from which a Cartesian coordinate representation (i.e. breaking them into scalar + bivector parts) can easily be derived. In this view the notion of “[mapping] complex numbers to geometry” is just tautological.

Maybe better would be to say that Euler’s formula shows how classical trigonometry (angle measure &c.) is what you get when you apply logarithmic thinking to Euclidean vector geometry.

If you see Geometry as flowing from something like Euclid's axioms, and (complex) numbers as flowing from a long line of advances in essentially bookkeeping/accounting, it is quite nice that they have such a neat synthesis.

A whole lot of the original bookkeeping/accounting was about adjudicating land boundary disputes and calculating land taxes based on area, or predicting/explaining astronomical events. In fact, the origins of both trigonometry and logarithms are in astronomy, though I’ll admit that the natural logarithm per se came from Bernoulli’s study of compound interest.

But yes, the connection between geometry and arithmetic is something people have been noticing for at least 5000 years, and is certainly important and very interesting. I just think this particular expression of that connection is oversold as being uniquely insightful or beautiful.

Sure. And if anything the more general equation e^(iphi)=cos phi + i cos phi is much more useful: it allowed me to forget all the special rules we learned in school about how to manipulate the trigonometric functions and just use my knowledge of exponentials instead.