Complex power of a complex number Can someone explain to me, step by step, how to calculate all infinite values of, say, 
$(1+i)^{3+4i}$? 
I know how to calculate the principal value, but not how to get all infinite values...and I'm not sure how to insert the portion that gives me the other infinity values.

Let us find in general $w^z$ where $w$ and $z$ are complex.  This expression is by definition equal to $$\exp\{z\ln w\}$$ where $\ln w$ is one of the complex logarithms of $w$.  That is, it is $w'$ where $$e^{w'} = w.$$  Suppose $w = re^{i\theta}$.  Then $$w' = \ln r + i\theta +2ik\pi$$ where $k$ is an arbitrary integer and the $\ln$ is the ordinary real-valued logarithm.  (Since $r\ge 0$ this is well-defined everywhere except for $r=0$, in which case we are dealing with $0^z$, which really is ambiguous.)
Putting this back into the original formula we have the answer, that $$\begin{align}
(re^{i\theta})^z & = \exp\{z (\ln r + i\theta + 2ik\pi)\}\tag{$\star$} \\
& =\exp\{z(\ln r + i\theta)\}\cdot \exp\{2ik\pi\cdot z\}
\end{align}$$
where $k$ is an integer.  

Now observe that although $(\star)$ seems to list an infinite number of solutions, they are not always distinct.  For example, when $z$ is a real integer,
the  second factor, $\exp\{2ik\pi\cdot z\}$ part is 1 for every choice of $k$, and so can be disregarded.
To find out how many values of $(\star)$ are distinct, one needs to ask about the values of $e^{2ik\pi \cdot z}$.  When $z$ has nonzero imaginary part,  or is a real irrational, these are all distinct and there are an infinite family of values of $w^z$, given by different choices of $k$.  But when $z$ is a real rational number with (lowest-terms) denominator $n$, there are exactly $n$ distinct values.
