It would appear that Google is preparing to migrate Google Search Console from /webmaster to /search-console URLs. This possible migration would be taking place about a year after the rebranding of Google Webmaster Tools to the Google search Console.
Found via this link (cc @jenstar @rustybrick) pic.twitter.com/i3vi0ZWbYk
— Dan Shure (@dan_shure) April 29, 2016
This is a picture from his tweet that shows him accessing https://search.google.com/search-console/mobile-friendly:
As of the writing of this post, I got the coming soon page:
On the plus side though, there was the new mobile-friendly testing page that he got:
Dan is currently part of a beta, and when he got a notification via the Google Search Console, the link in the notification lead him to the new URL.