Separate imaginary and real parts from complex expression

I learned about complex numbers after I was trying to create a fractal object.

The main problem is that I have an equation with complex numbers and I have to separate their parts (real & imaginary) to calculate the next iteration.

Some equations like the $f(p) = p^2 + c$ are obviously and easy to solve them. But some of them are using the exponential form of the complex numbers, that really bothers me.

Take for example this equation:
$$\begin{equation*} F(P) = c*e^{-p} + p*p~\text{Where}~c = u + v*i~\text{and}~p = a + b*i. \end{equation*}$$
I solved it partly by this way - express $e^{-p}$ like $1 / (e^a \ast (\cos(b) + \sin(b) * i))$ and so on... But in the end I couldn't see the relation to separate the groups.

Any help and advice how to proceed in this or similar case will be greatly appreciated.