Command Chat
This idea specifically pertains to command driven channels, like Twitch Plays Pokemon, SaltyBet, and the like. Generally in channels like this it becomes very difficult to read conversation when the chat is flooded by commands, and depending on the channel, bot responses to those commands.
My suggestion would be to grant channel owners the option to have a bot/command specific chat log, perhaps with a simple vertical separation from the standard chat, so as to separate commands from conversation.

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LomariFairchild commented
Yes please! I have a fun widget that shows information from the game but absolutely fills my chat with commands. Hiding them would be so nice!!
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Sparx07 commented
My suggestion is per-message filtering for the Chat Box based on the first character of the message.
While there is a way to mute users/bots by account name, many chat bots are interactive; the channel owner, moderators and chatters can send commands to the bot through the chat.
For example, a command may be “!ChangeGame Rocket League”, triggering the bot to change the game title to “Rocket League” on Twitch.
In this case, if ‘!’ character could be specified as a command character in the settings and filtered as to not show up in the widget, it would allow streamers to hide bot & bot-related spam from showing up on the stream.