Assume $O$ is the compliment of the non-positive part of the real line to the complex plane. This is an open and connected set. Only one of the values of $\sqrt z$ in $O$ has positive real part. With this choice $w=\sqrt z$ has unique value in $O$.
Attempt: $$a^{\frac12}=e^{\frac12\log a}=e^{\frac12(\ln |a|+i\arg a)}=e^{\frac12\ln|a|}\frac12(\cos\arg a+i\sin\arg a)$$
Then the real part is $\frac12e^{\frac12\ln a}\cos\arg a$. Since cosine is $2\pi$ periodic, regardless of the choice of the argument which takes values from $\{\arg_0+ 2k\pi:k\in\mathbb Z \}$, we get a unique value of the square root, do we not?
No, you didn't distribute the $\frac{1}{2}$ in the exponent properly. It should be $$e^{\frac{1}{2} \ln |a|}(\cos (\frac{1}{2}\arg{a}) + i \sin (\frac{1}{2}\arg{a})).$$ Now you can see that the choice of arg is important.