Having seen this as a talk I am very very happy to see it as a post that can be referred to. 🎉
UPDATE: March 29 2016. Twitter have announced that they will support a limited amount of alt text on images posted to twitter. Unfortunately they will not be enforcing alt text or using AI to determine what is in an image.
I just switched to using a command-line Twitter client and I was again reminded about how limiting it is to use Twitter when you cannot see the images contained in a tweet.