May I offer the four-vector notation as an example from physics? Quoting Feynman:
The notation for four-vectors is different than it is for three- vectors. [...] We write $p_\mu$ for the four-vector, and $\mu$ stands for the four possible directions $t$, $x$, $y$, or $z$. We could, of course, use any notation we want; do not laugh at notations; invent them, they are powerful. In fact, mathematics is, to a large extent, invention of better notations. The whole idea of a four- vector, in fact, is an improvement in notation so that the transformations can be remembered easily.
The Feynman Lecture on Physics, Volume 1, Chapter 17.
A trigonometric notation that Feynman invented in his youth did not catch on.