Separate colors for each chat line, not user.
Change the color preference in chat, which shows the username in a single static color to be a color per chat line.
This means that there will be 4 colorblind and dark theme color friendly colors that appear for each username per chat line.
User 1 (in purple): blah blah blah
User 2 (in blue): blah blah blah
User 1 (in orange): blah blah blah
User 3 (in white or dark grey: blah blah blah

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Crev commented
Love this idea, if you need something in the interim highly suggest using BTTV until Twitch can implement this.
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BilkyAsko commented
Having just watched a replay of a stream, it was extremely difficult to follow the chat, which mostly consisted of people whose usernames were coloured in difficult to read shades.
I could not find an option to set the chat to readable colours. At the very least, this should carry over from the normal chat settings, but it should definitely be an option to set this on chat replays.