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How performs the function $ z^{1/2}$ in the complex plane? Thanks for your help I know it's a multi-valued function and that we must be careful with the branch on which it is defined, one of my questions is what is the main branch?, Where is continuous and which is defined?

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    I just want to see their behavior in two dimensions2012-05-21

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It is defined on $\mathbb{C}\backslash{(-\infty, 0]}$ (which is the main branch of the complex Logarithm) and is continuous everywhere.

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    I'm more familiar with the convention that it is defined everywhere but discontinuous on the negative real axis (which gets sent to the positive imaginary axis), as in lhf's answer.2012-05-21
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The principal square root function uses the nonpositive real axis as a branch cut (Wikipedia).