To be clear Feynman only used his own notation when he was a teenager. He soon realised that having others understand your work was important and switched to conventional notation.
> Is it another level of coolness for someone who understands math (which is cryptic enough by itself)?

So much crankery hides behind "Oh what I'm doing is so revolutionary that it can't be described using standard language/symbols". If you genuinely want others to understand and/or accept your work, then you'll make an effort to describe it in ways that engage others (and possibly lead them to eventually understand why you've needed to use non-standard language/symbols), rather than expecting them to put in effort to learn something which may not be worth the payoff. You might be able to get away not doing this if you've already non-controversially demonstrated your competence, but even then, you might still only be given a limited amount of rope.
Math is as much a language as it is an art and a science. Sure, you can invent your own language and it might make things simpler (ever tried to confess your love in French and German? Then you know what I mean!). But it also means that you become harder to understand and will need a translator.

When I'm working on a problem, I use plenty of short-hands for all kind of stuff. I invent symbols for recurring things left and right. It does make thinking easier when I have less clutter on my paper. But once I'm finished and it's time to write things down, then all those symbols and short-hands go away.... unless they turned out to be useful, but then they get properly introduced first.
Iverson Notation maybe? Ken Iverson invented a notation for his math classes that got rid of many of the inconsistencies of modern math notation and, most importantly, is linear. It is not too difficult to learn and I really like it. A few ideas of his made it into modern math, but unfortunately, most of it kinda got lost, except in the programming language APL, which is a direct decendant of Iverson Notation.
Lesniewski invented a logical notation which was entirely his own.  The weirdest-looking bit was propositional connectives as spoked wheels that encoded the truth table in the spoke pattern.  His work was not crankish and not entirely trivial but didn't achieve enough for him to persuade anyone else to go for it.
I've used it once in the realm of digital logic to express a multiplexor block

z=mux[x]{1—>y1, 0—>y2}

Represents a multiplexor controlled by x, z equals y1 if x=1, and y2 if x=0

I find it to be clear enough for anyone to be able to parse it
I tend to use extremely strange notation, having learned formal logic and set theory from logicians before embarking upon maths. It's led to more than a bit of confusion between me and others.
I think using your own symbolism in the way you described it's a infatuated way of saying: ,, I'm smarter than you". I mean, what's the point in using another notation for sine ? Beauty in math consists in it's universality: any individual with a enough mathematical maturity can understand it via his intuition.
I mean, it's not really what you're talking about but i think it's quite common for mathematicians to invent new definitions for things. And if your definitions are sufficiently new, there won't be any good notation for what you're working with so you have to invent some.

I spend an unreasonable amount of time trying to think up notation and names for things that will be easily understood by as many people as possible. Lol i certainly don't feel "cool" for having cryptic math. I often have a bunch of convoluted notation that basically only makes sense to me and i usually feel like a failure if i can't rework it into something my advisor can easily understand.

A lot of the time my lack of good notation comes from ignorance on my part btw. Because it turns out that what I'm describing is actually this other thing I didn't learn about yet.
I don't do this much. Perhaps using sigma, gamma, and tau for the trigonometric functions; or xi for exponential, these are only time and space saving measures however. Something more useful might be splitting up the concept of equal sign into a number of distinct notions such as 'set this X to 3' (a la 'Assuming x = 3') versus 'this X must be 3' ('Thus, x = 3'). They aren't actually different semantically but they function pragmatically for you as hints.

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