I have read a paper which has stated the following:
A conformal image of a Brownian motion is a time changed Brownian motion.
The paper cites R. Durret, Brownian motion and martingales in analysis, Wadsworth Advanced Books & Software, 1984. However, I don't have access to that book, so I was trying to figure it out myself. As I have minor understanding of conformal maps, I am left clueless with this problem. Hints?