Currently Claude.ai doesn't include a means for exporting a conversation. This add-on to Chrome and Firefox captures the JSON page content as it's downloaded, converts it to HTML that closely mimics the behavior of Claude in the browser, and downloads the result. The basic structure of the add-on was written using Claude.
Firefox and Chrome have different requirements for extensions, so there are two implementations, each in a separate directory.
To install in Firefox, download the claudeExport.xpi file to your computer. In the add-ons window, click the settings gear and then click "Install add-on from file". Browse to wherever you downloaded the .xpi file and select it. Add-ons that have been checked by Firefox contain a META-INF directory which contains a signature generated by Mozilla. This is an automated check so while it should provide some confidence, it's not a guarantee. Of course, you can look at the source code in the repo.
To install in chrome, open the debugger interface by typing "chrome://extensions". In the upper right corner enable "Developer Mode". Then click "Load unpacked". Navigate to the Chrome folder and click on manifest.json. You may be tempted to "pack" the extension into a .crx file so that you don't have to use developer mode. This won't work. Chrome will only install .crx files from the Chrome store. Getting stuff into the Chrome store is hardere than the Firefox store, so I didn't bother.
Because the extenson captures the JSON as it's downloaded, if the extension is not active when the page was initially downloaded you will get a message "no conversation captured" when you try to view or download the captured conversations. If this occurs, just reload the page and try again.
By default, printing does not include non-inlined artifacts - ie, artifacts that appear in the slide-out panel. There is a checkbox labeled Print All Artifacts in the pop-up menu. If you check this box before exporting the HTML, a all the artiprinted version will have all artifacts printed where they occur in the conversation. The browser view of the file is the same whether or not the box is ticked.