Is it possible to calculate something like ?
I recall reading somewhere that is calculable (I think its a real number), so I was pondering whether or not this was true for complex numbers in general?
Yes you can. It's no simple task, but it requries you write the complex number in its exponential-polar form.
Complex Exponentiation -- from Wolfram MathWorld
Natural Logarithm -- from Wolfram MathWorld
Middle way down.