I played along...
...with the call from "windows support" having "demonstrated" that "Oh my god your machine is badly infected with a polymorphic virus" (on the basis that there were errors listed in the windows error log file).
Next they sent me to logmein.com (which, shamefully, doesn't give any security warning that naive users would understand) so a "technician" could investigate the problem.
Obviously that's where I stopped but looking around the web, there's people who've been scammed out of $200 - that makes an annual salary from this scam of $50,000 if they get one "success" a day.