Planck's constant $h$ came out of his considerations of black body radiation and features prominently in quantum physics. Recently I came across the statement that $-i\hbar = pq - qp$ for elements $p$ and $q$ of a rather general non-commutative ring, but I it was not quite clear how "general".
Now I wonder: is the numerical value of $\hbar$ more something like $\pi$ or Euler's constant $e$, which come out of "pure math" (I now this is a bit fuzzy) or is it more like the speed of light, $c$, which can change its numerical value depending on which fundamental units one is working with?
If $\hbar$ can be derived in non-commutative algebra without relying on physics input, I would be grateful to get a link to a derivation.
Planck's constant is a dimensionful quantity; its numerical value depends on which units you measure it in. Since the units are more or less arbitrary conventions, the precise numerical value is of no particular mathematical significance, nor does it have deep physical importance.
In particular, one of the arbitrary unit choices that go into determining the value of Planck's constant in SI units is the kilogram, which is still defined as the mass of one concrete physical object stored at the BIPM. If the kilogram definition is revised to be a matter of objective physical phenomena, it is conceivable that the value of Planck's constant would in theory be a deductive consequence of a sufficiently powerful Theory Of Everything. However, all the currently proposed such theories still seem to involve aribtrary constants, some of which are dimensionless.
One can choose to work in natural units which corresponds to defining that Planck's constant and a bunch of other physical constants considered fundamental, have nice exact numerical values such as $1$ or $2\pi$. However, that is still just a convention, and the particular value the constants take in such natural units is a product of the unit choice rather than a separately meaningful mathematical thing.