According to Cauchy's integral theorem, a holomorphic function in a simply connected domain is path independent. According to Morera's theorem, a path independent and continuous function is holomorphic. In other words, each one implies the other.
But that begs the question, is there a way, some theorem, to find either easily? In the real set, the curl operator can be used to find path independence, for example.
Yes, a differentiable complex function $f$ of a complex variable is holomorphic if and only if the Cauchy-Riemann equations hold. Those are totally analogous to the vanishing of curl.