The only reference I'm familiar with that deals with arc accessibility in the plane is Wilder's Topology of Manifolds, which is getting close to 100 years old. There is a book on Cluster Sets which covers it up to what's typically needed in complex analysis; there's also a monograph on some of Bell's work, but it's not really didactic. There have been plenty of refinements and applications of accessibility since the Wilder book in particular, but I had trouble finding a serious treatment for topologists, rather than complex analysts.
Anyone know a good one?