Customizing the end of the chat experience would be great - and it would be nice to have a couple of options for this, as well, rather than just one version for all chats.
The “Your chat has ended” box that appears at the end of the chat reads a bit too strong, to me.
An alternative solution might be to simply ask the user if they want to end the conversation, like @dtempleton mentioned, or you could just put one line of small text at the end of the conversation stating that the chat has ended, and a link with ‘continue chatting’ or ‘start a new chat’ after. Placing it in a larger box with extra text and a button seems intrusive.
An additional issue with the box is that it covers the last one or two speech blubs in the conversation, so it’s harder to finish reading or click on links that were sent, etc.