I've been programming for a very long time (yes I remember a time before PC's) and have a lot of different languages that I have worked on.
Each language tends to borrow from the ones before and for that reason alone knowing your way around a few of the "root" languages can give you insight into why things are the way they are (uncluttered by more modern methods), in the same way as fixing a Model-T will tell you more about engines than fixing a Ferrari.
However, while my knowledge of assembly language gives me the odd clue as to why some things work the way the do (or more often why they don't work), would I learn assembly language now if I was starting out again? No, my time is more valuable than that.
And while C is more common than assembly language, the same applies. Unless you plan to work in C or have a lot of spare time, I would say your time would be better spent in more modern languages. The more of them you know, the more you will "see" the common roots, with the added benefit of making yourself more marketable in the current workplace.
Just the opinion of an old warhorse.