Zendesk could develop a mobile app for the chat function. This way, the person who has the app installed could receive and replay to messages from customers, without being logged in to the support platform on the computer.
Other providers, such as Intercom, offer this function already. The chat is available 24/7 for the customer - messages are forwarded to the mobile app if the agent is not online on the platform.
It is especially useful to startups and other small companies, besides giving more mobility to agents.