In the way that we got rid of the rieman integral for most purposes (not including complex analysis) with the lesbeague integral, which is much more "clean", is there a way to get rid of parameterizations in complex analysis? I know that the integrals there are independent of it, as is rieman integral via rieman sums.
Another motivation for this, is that it feels "ungeneral" to have to deal with domains that have a boundery is pathwise smooth, wouldn't it be much better if could talk complex analysis on arbitary open connected sets?