Only problem I see with this is making things slightly more confusing.
The in game time we walk about here would be a "FH" time and not an real life time anywhere in which chats that do have time after messages like the one suggested usually goes by, some user might write something one time, and another write something just slightly after and all the sudden it's 20 minutes after even though there was just 1 minute real time that passed by.
If there was a clock shown at all time unless disabled, it wouldn't bring the same "oddness" since everyone would have the actual clock in the corner, the icon displaying where on the day but also the actual day/night cycle inside of the game.
Having a time stamp in the chat would also loose it's own point slightly cause
A. It wouldn't be showing the actual ingame time anywhere and you had to write in the chat to see what the game time is.
B. We wouldn't even be able to use for easier moderation of the game like real time chats using this can, cause it doesn't even show the "actual" time.
That chatbox is kinda flooded as it is right now if you'd ask me as well.