When considered as a function defined on the complex plane, the exponential function retains the important properties
\!\, e^{z + w} = e^z e^w
\!\, e^0 = 1
\!\, e^z \ne 0
\!\, {d \over dz} e^z = e^z
for all z and w.
It is a holomorphic function which is periodic with imaginary period 2pi and can be written as
\!\, e^{a + bi} = e^a (\cos b + i \sin b)
where a and b are real values. This formula connects the exponential function with the trigonometric functions and to the hyperbolic functions. Thus we see that all elementary functions except for the polynomials spring from the exponential function in one way or another.
See also Euler's formula.
Extending the natural logarithm to complex arguments yields a multi-valued function, ln(z). We can then define a more general exponentiation:
\!\, z^w = e^{w \ln z}
for all complex numbers z and w. This is also a multi-valued function. The above stated exponential laws remain true if interpreted properly as statements about multi-valued functions.
The exponential function maps any line in the complex plane to a logarithmic spiral in the complex plane with the center at the origin. This can be seen by noting that the case of a line parallel with the real or imaginary axis maps to a line or circle.
what are you talking about?