FYI, Mirroring and casting are two different ways of connecting your phone to your TV.
But to answer your questions chromecast works great with the xperia z3. chromecast will play apps I.E Netflix, YouTube via wifi to your TV using a network adapter type device plugged into your hdmi port on your tv. If you have a smart tv you can pretty much mirror what's on your phones screen directly.
Sent from my Sony Xperia Z3 via the Coconut Wireless \000/