Why can't the exponent laws be extended to complex numbers?

Question

$(x^a)^b=(x^b)^a=x^{ab}$ only when real numbers are involved. This implies that the base must be a positive number. For example, with $x=-1$, $a=2$, and $b=\frac{1}{2}$, $(x^a)^b=\{(-1)^2\}^\frac{1}{2}=1$; but with $x=-1$, $a=\frac{1}{2}$, and $b=2$, $(x^a)^b=\{(-1)^\frac{1}{2}\}^2=-1$. So the rule breaks when a non-real number is involved ($(-1)^\frac{1}{2}$). The question may sound a bit weird, but why is it so? And is there a special set of exponent laws that work for the complex numbers (as well as real numbers, as a subset)?