Is it possible to get desktop notification about each chat using your Live Chat plugin?
Yes, it is possible. You have to add Live Chat extension to your Chrome or Chromium browser.
Open Chrome menu button in the upper right corner of browser–> More tools –>Extensions–>Get more extensions –> in chrome web store search box type “wp-chat” –> choose “Chat by wpadm.com” –> Add to Chrome.
Once the chat plugin is added, login with email and pass from the account you want to receive and answer messages at.