Welcome to the forum.
Probably any flash with a standard ISO hot shoe will at least fire from the K-1.
The Pentax AF201FG flash should be a lot cheaper than the AF 360 FGZ II. And there are lots of third party flashes available that will work with the K-1 and support Pentax's P-TTL flash system, from Yongnuo (YN585EX), Shanny (SN600FGZ), Metz (26AF-2, M400, 44 AF-2, 52-AF1, 64 AF-1) etc.
Search B&H for 'Pentax flash', look at brands like Metz, Pentax and Sigma, and you will see that there is a decent range available.