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For the cartesian plane, or other planes such as polar, points are plotted by a function. One value of x coresponds to a value of y. (or r to theta, or whatever.)

The complex plane isn't a plot of functions, just of a single number, it's imaginary part and it's real part.

Is the complex plane just a choice of how we want to graphically represent a complex number? Or does it have more innate meaning?

Do identities such as euler's completely rely on this graphical representation of a complex number to make sense?