Why is the principal square root of a complex number not well-defined?

[PFuser1232]

Within the context of real numbers, the square root function is well-defined; that is, the function $f$ defined by:
$f(x) = \sqrt{x}$
Refers to the principal root of any real number x.
Is it true that this is not the case when dealing with complex numbers? Does $\sqrt{z}$, where $z ∈ ℂ$, represent more than one value?

 

[DivergentSpectrum]

It depends more on how you define nth root, not so much whether the input is a complex number.
If you define ⁿ√x as the inverse function of xⁿ, then yes, there is more than one value, and infact more than one inverse function.
more reading youll enjoy:
http://en.wikipedia.org/wiki/Root_of_unity

 

[PeroK]

There is, in fact, a definition of the principal square root of a complex number:

http://en.wikipedia.org/wiki/Square_root#Square_roots_of_negative_and_complex_numbers