You need to basically join the table to itself.
You can perform a query like this:
The Query
SELECT chat.chat_id, chat.chat_user AS chat_user1, chat2.chat_user AS chat_user2
FROM chat as chat, chat as chat2
WHERE chat.chat_id = chat2.chat_id
AND chat.chat_user != chat2.chat_user
GROUP BY chat_id
Which will join it to itself and check for all groups of chat_id and return the result with an extra column as you want.
SELECT AS
Importantly to change column names you can SELECT column1 AS new_column1, column2 AS new_column2
This is particularly useful if you have ambiguous column names which of course you will when joining a table to itself in essence.
WHERE / AND
Here we are first checking the chat_id is the same so all chat_ids that equal 14 for example get returned, however to stop it creating multiple rows we tell it to ignore chat_users that have already been put into chat_user2 on the first run through of the code.
GROUP BY
This stops it creating duplicate rows of data and returns one result for each chat_id
I've assumed you're table is called chat for this. If not replace chat with your table name