Referring to precisely the comments on the original question posted here, one user says,

I suppose you are talking about exponentiation for complex numbers, rather than for reals. Also, isn't it the case that $a^b$ is not unique for any $b$ that is not an integer (assuming we're working with complex numbers)?

And it seems like the resolution was that this is true for complex $a$ and $b$, but I didn't think this was correct. I thought exponentiation defined for a base $a \in \mathbb{C}$ and exponent $b \in \mathbb{C}$, that $a^b$ must be unique. Am I misinterpreting the discussion or is there something that I'm missing?