I am comparing two ways of defining the integral along a path of a function of complex domain and value. One is the one given in Conway's Functions of one Complex Variable and the other is given in Stein's Complex Analysis. The one given by Conway proceeds in the fashion of the Riemann-Stieltjes manner: first begins by defining functions of bounded variation of a complex value with domain a compact interval of the reals (which he calls "Rectifiable Curve") and then proceeds to prove that every continuous function with domain a compact interval is integrable with respect to a rectifiable curve.
Stein's approach is simpler: he defines the integral of a continuous function $f: G \subset \mathbb{C} \to \mathbb{C}$ along a continuously differentiable curve $\gamma : [a,b] \to G$ to be
$\int_{\gamma} f = \int_{a}^{b} f(\gamma(t)) \gamma'(t) dt$
(Here I was forced to assume that if $f:[a,b] \to \mathbb{C}$ is a function such that $Re(f)$ and $Im(f)$ are Riemann integrable, then we define $\int_{a}^b f = \int_{a}^b Re(f) + i \int_a^b Im(f)$)
I get that Conway's approach is more general to Stein's in the sense that the deffinition of Stein is just the particular case of Conway's deffinition with the restrictions stated above.
What are the advantages of the definition by Conway with respect to Stein's? One is that it is more general, but this advantage seems to general for me, so, I want to know in which particular cases is the definition given by Conway draws advantage to Stein's.
Also, when I had a course involving the Riemann Integral in the reals, the Riemann-Stieltjes approach was also used. Then, a year later, I began reading Bartle's A modern theory of integration, in which the Kurzweil-Henstock construction of the generalized Riemann integral is given and, this approach is more general and simpler than the Riemann-Stieltjes, so I am wondering if it would be beneficial to give a Kurzweil-Henstock approach to the line integral in $\mathbb{C}$, and if so, why isn't any book (at least of my knowledge, correct me if there exists one in some text) giving this construction?