How about nonassociative rings? Personally,I've always had a problem calling something where the multiplicative structure is nonassociative a ring.I think a lot of algebracists do too,becuase unless the author is a researcher in ring theory,they're never mentioned in the standard algebra texts at either the undergraduate or graduate level. The only standard books that mention them? Herstien and Jacobson,Both promenient ring theorists. Anyway,the quarterions form a nonassociative ring over C,I believe-the Lie and Jordan rings are the prototype examples.
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