As a math graduate, octonions (and quaternions) at least to our group, was something that got discussed once or twice and not much after that. However, I would not call them weird or unusual, at least not more weird than something like a near-ring which similarly simply drops one of the common assumptions for rings (I'm not sure which, actually). As mentioned in some of the comments, nonassociative fields do get studied. Studying not necessarily commutative structures such as (all) groups is in fact an even much more common thing to do.
I would say that studying octonions exclusively would be something I personally would avoid, as I would rather try to study the four structures (reals to octnonions) together, either more generally (e.g., group theory and ring theory) or more abstractly (e.g., as members of categories) and form an opinion on whether I think octonions in particular are useful for the questions that I want to ask.
That is not to say that the research here is not interesting, but it is difficult to judge that from "popular" mathematics articles. I got the impression that the author of the article places a much higher priority on the pictures accompanying the post.
I remember that I found it interesting that studying the four dimensional spacetime bears more fruit than stopping at three space dimensions, and that at the same time from complex numbers the next structure ends up also having four dimensions (i.e., being modelled by 4-tuples). However, apart from being interesting in this narrow sense, I do not know whether this suggests any creative yet precise mathematical questions.
I would say that studying octonions exclusively would be something I personally would avoid, as I would rather try to study the four structures (reals to octnonions) together, either more generally (e.g., group theory and ring theory) or more abstractly (e.g., as members of categories) and form an opinion on whether I think octonions in particular are useful for the questions that I want to ask.
That is not to say that the research here is not interesting, but it is difficult to judge that from "popular" mathematics articles. I got the impression that the author of the article places a much higher priority on the pictures accompanying the post.
I remember that I found it interesting that studying the four dimensional spacetime bears more fruit than stopping at three space dimensions, and that at the same time from complex numbers the next structure ends up also having four dimensions (i.e., being modelled by 4-tuples). However, apart from being interesting in this narrow sense, I do not know whether this suggests any creative yet precise mathematical questions.