If one accepts radian as a fundamental unit, does it make sense that action and angular momentum have units differing in radian to the power of one?

The same question applies for energy and torque.

The origin of this is that, we usually write

$h\nu = \hbar\omega$

and also $\nu$ has units of $1/Time$, and $\omega$ has units of Angle/Time.

Interestingly, if so, that will make $\hbar$ and $h$ have different units. The first being the fundamental unit of angular momentum and the second being the fundamental unit of mechanical action.

• At the end it could be that question boils down to whether such distinction is useful (and not self-contradictory)?

Although I don't write it explicitly it usually helps me think that these pairs of quantities have different units.