If you're taking suggestions, I think a three-column format would be cool. Basically, the first column is the thin icons you have now. The second column is the tweet column. The third column (equal in size to the second column) would appear when you click on a tweet; instead of loading the responses to that tweet as a modal view as it does now, the responses would appear in the right column.

Basically, this would make Twitter a lot more like several email clients or an RSS reader. Quick and dirty mockup: https://i.imgur.com/Uh9Rbcz.png

Also, would be nice to force Twitter into "latest tweets" mode, and also to provide a userscript for those of us who prefer to avoid untrusted addons.