Write i^i in the form u+iv
Write $i^i$ in the form $u+iv$.
Explanation of Method
$i=e^{i(\pi/2+2k\pi)}$ for $k\in\mathbb Z$, so $i^i=e^{i\cdot\text{Log}(i)}$ is multi-valued; the exponent $i\cdot i=-1$ turns every branch into a positive real number.
Step 1 · General logarithm of $i$
$$ \text{Log}(i)=\ln1+i\left(\frac\pi2+2k\pi\right)=i\left(\frac\pi2+2k\pi\right) $$
Step 2 · Raise to the power $i$
$$ i^i=e^{i\cdot i(\frac\pi2+2k\pi)}=e^{-(\frac\pi2+2k\pi)} $$
Step 3 · Principal value ($k=0$)
$$ \boxed{i^i\big|_{k=0}=e^{-\pi/2}\approx0.207880} $$
Step 4 · Other branches
$$ k=1:\ \ i^i=e^{-5\pi/2}\approx0.000388, \qquad k=-1:\ \ i^i=e^{3\pi/2}\approx111.318 $$
Remarkable property
Although both the base $i$ and the exponent $i$ are purely imaginary, every value of $i^i$ is a positive real number, forming the geometric sequence $e^{-\pi/2}\cdot(e^{-2\pi})^k$.
Result
One of the classic surprises of complex exponentiation: an "imaginary raised to an imaginary power" is not only real but positive, for every branch.