Analytic continuation of $z^{1/2}$

Question

Is there a simple argument to show that a branch of $z^{1/2}$ can be analytically continued along any curve beginning in its domain? I can show this for the circle using power series, or by defining $f_t(z) = e^{it} f(e^{-2it} z)$ for $t\in [0,2\pi]$. But is there a proof that works for a more general curve?