If you realize where e comes from, which is compounding and what we call an exponential process, being that "the current rate of change is proportional to the present value", then the exponential of a complex number is no different to the exponent of a real number. In the case of one-dimensional real numbers you assign the slope of the function to the current value. The complex case is exactly the same, but if you think instead in the two-dimensional Argand plane and complex algebra, the slope is the tangent to a circle proportional to the angle at that radius (which curve is the one that at every point the slope is equal to its complex value?).

Therefore with the rules of complex arithmetic the tangent provides the "curling" effect in the curvature of a periodic circle, while in the real case you get the common compounding shape of the exponential curve. The relation to sine and cosine is just the projection into direct(real)/quadrature(imaginary) components in either fixed or intrinsic coordinates. Same when you expand the exponential into its complex power series.

BTW, this is actually the essence of infinitesimal transformations in continuous groups and the exponential map, which generalizes this concept to other types of numbers or abstract objects (i.e. Lie group theory).