Various emperors renamed months after themselves, not all renames persisted. But they didn't include new ones.

The Roman Senate changed the beginning of the legislative year at some point to January 1, long before the republic was gone.

But this doesn't mean it was consistent ever after in medieval Europe even inside what we call Germany now.

This was only changed from the 16th century on. see New_Year#Adoptions_of_January_1

I've been sometimes confronted with the church-year of the Catholic Church which starts with the first advent.

I really also like the Ottoman calendar depicted page here with all the various details for all ethnicity.

Reminds me of a Turkish friend who was so annoyed about the red-tape he faced when studying in the UK, that he told the authorities that is father was born 1337.