According to WolframAlpha, ii=e−π/2 but I don’t know how I can prove it.
Here’s a proof that I absolutely do not believe: take its complex conjugate, which is (ˉi)ˉi=(1/i)−i=ii. Since complex conjugation leaves it fixed, it’s real!
EDIT: In answer to @Isaac’s comment, I think that to justify the formula above, you have to go through exactly the same arguments that most of the other answerers did. For complex numbers u and v, we define uv=exp(vlogu). Now, the exponential and the logarithm are defined by series with all real coefficients; alternatively you can say that they are analytic, sending reals to reals. Thus ¯expu=exp(ˉu) and ¯log(u)=logˉu. The result follows, always sweeping under the rug the fact that the logarithm is not well defined.