I saw this in Der treffende Ausdruck:
Die Nachbarin grüßte ihn kurz und ging ihres Weges, aber der Kollege unterhielt sich mit ihm, während sie auf den Bus warteten.
I’ve never seen a gehen plus genitive construction before. From the context I assume it just means she went on her way but I’m curious as to why one would use the genitive there, if it sets a certain tone, and if using gehen + genitive is appropriate/necessary in some contexts.
Congratulations, you have stumbled upon an idiomatic expression. Check the Duden page for Weg to find a mention of it under Wendungen, Redensarten, Sprichwörter.
Seines Weges gehen is a fixed expression meaning to physically walk along one’s ways. It should not be confused with seinen Weg gehen which can also be used in the metaphoric sense of sticking to one’s path.
I am not aware of any other constructions that use gehen plus genitive. However, it is paralleled in ‘Wohin des Weges?’, a now frozen expression for ‘where are you going?’
