How do I purge Pantheon desktop?/Why does this window appear?

ppa-purge is unneffective with that ppa, (elementary-os/daily) & the same with autoremove.

Open Synaptic,

click Settings, Repositories

Click Add and add back the ppa you used previously & update sources.

Click the Reload button in the upper left corner, then Origin. Click on the entry for that ppa/now similar (entry highlighted above). With shift+click highlight all the packages & mark for removal, and remove them.

Then remove the PPA from your software sources Under Settings, Repositories.

Source: comment from @doug

The clue here is "captive-login" in your last screenshot. This is a new feature [1] in Elementary OS which seeks to make connecting to the Internet seamless when behind captive portals (hence the name). It is similar in nature to features on mobile platforms like iOS and Android. When on a restricted connection such as hotel or airport wifi a check is made to see if the computer can reach the 'real' Internet, and not the blocked off bit you have to sign into, in order to gain access.

So to me this just looks like a remnant of that feature kicking around, and needs removal.

[1] https://code.launchpad.net/~spinatelli/elementaryos/captive-login

Run

sudo apt-get remove --purge capnet-assist