Alternatively if you don't like coordinates, you may notice that multiplication by a unit complex number is linear and preserves the complex norm (i.e. modulus), so it defines an orthogonal transformation of the plane.
Topologically, unit complex numbers are the unit circle which is connected, and 1 is part of this circle so all of these transformations must have determinant 1: these are rotations.
Topologically, unit complex numbers are the unit circle which is connected, and 1 is part of this circle so all of these transformations must have determinant 1: these are rotations.
No Euler formula.