Introduce arg, Im, Re (or maybe im, re). So, in particular, write things like:

arg(z) = atan2(im(z), re(z))

and not

arg(x+iy) = atan2(y, x)

because the latter makes things more complicated. It turns out everything can be written using arg, re and im.

Pick a branch cut, at negative x-axis for arg. Everything else follows from it. Few examples valid for all complex values of all variables:

exp(log(z)) = z

log(exp(z)) = z + 2*pi*i*floor( (pi-Im(z))/(2*pi) )

exp(z)^k = exp(k*z) * exp(2*pi*i*k*floor( (pi-Im(z))/(2*pi) ))

log(z1 * z2) = log(z1) + log(z2) + 2*pi*i*floor( (pi-arg(z1)-arg(z2))/(2*pi) )

exp(x+y) = exp(x)*exp(y)

z^(x+y) = z^x * z^y

(x*y)^a = x^a * y^b * exp(2*pi*i*a*floor( (pi-arg(x)-arg(y))/(2*pi) ))

(z^a)^b = z^(a*b) * exp(2*pi*i*b*floor( (pi-Im(a*log(z)))/(2*pi) ))

Then add some worked out examples:

0 = log(1) = log((-1)*(-1)) = log(-1) + log(-1) + 2*pi*i*floor( (pi-pi-pi)/(2*pi) ) = i*pi + i*pi + 2*pi*i*(-1) = 0

One can find more here:

http://functions.wolfram.com/ElementaryFunctions/Log/16/ShowAll.html

Usually one has to be extremely careful and think about all the branches etc. But the above approach provides a robust, simple, algorithmic way to work with complex expressions. That should be stressed.