I have to say that the new mobile chat is pretty freaking good. I wouldn't change anything as far as usability goes.
However, it doesn't seem to have an audible ping cue. When someone replies to me, I don't hear any noise. Perhaps this is by design, or maybe there are limitations on mobile browsers. I wouldn't mind a push notification, if it is necessary for this feature, as long as I can turn it off.
I would like it to ping at me if the browser has a page open in chat and if I am currently not on that page, or the screen is off.
Stock and updated Chrome Browser, Samsung Galaxy S5, verizon, USA.