If you download your Twitter archive it arrives wrapped as a static HTML page, which is not very useful for doing anything with, and worse: it requires the original account to be still active to do useful things like enlarge the images since they use t.co links.

So here's a Python script to convert a Twitter archive to markdown or other formats: https://github.com/timhutton/twitter-archive-parser

Now you can archive your tweets in any way you want.