my favourite intuition about Euler, which is not really an explanation, somewhat tautological, and may or may not be wildly incorrect, but I like it nonetheless:
e^x is a function whose value is its rate of change. (De^x=e^x). Now imagine the unit circle by taking a point an unit away from O, and set "rate of change" perpendicular to that vector. You will end up with Df(x) = i f(x), which really only works when f(x) = e^ix, supposing i means perpendicularity.
In the real plane you get exponential growth: the rate of change is equal to the current value.
In the Argand plane you get a curling action due to the quadrature effect of the imaginary unit. The rate of change at each point is the tangent, and the result is therefore a circle.
Lie infinitesimal displacements capture this nicely, and also render the generic case which is a similarity transformation, e.g. rotation through two half reflections, e^(-w/2) * x * e^(w/2) like those found in quaternions and Clifford algebras.
e^x is a function whose value is its rate of change. (De^x=e^x). Now imagine the unit circle by taking a point an unit away from O, and set "rate of change" perpendicular to that vector. You will end up with Df(x) = i f(x), which really only works when f(x) = e^ix, supposing i means perpendicularity.