Which is the most fundamental constant between the Planck constant $h$ and the reduced Planck constant $\hbar$? This question is related to Planck units (also called natural units, absolute units or God's units).
I'm wondering which constant is the most fundamental and should be normalized to 1. I would like as much explanation as possible please.
 A: Generally $\hbar$ is normalised to $1$ There is no reason why you cannot normalise $h$, but $\hbar$ is the convention for similar reasons to why you define $\hbar=\frac{h}{2\pi}$ in the first place; it gets rid of lots of annoying factors of $2\pi$. Here is a wikipedia article on the subject, which rightly points out that there are a number of different normailsation systems you could use. This is just the most widely used one.
A: I would say that $\hbar$ is the more fundamental constant, because it has a mechanical interpretation: it is the size of one "lump" of angular momentum. We have $h$ because quantization of angular momentum wasn't understood for the first couple decades of the 20th century; $h$ converts between frequency (in particular units) and energy (in particular units). But you don't need a particular system of units to measure $\hbar$.
A: The question has two aspects: 


*

*Which constant is more fundamental?

*Which constant should be set to 1? 


@rob and @nivag have given excellent answers to the question in the title, and those answers are deserving of upvotes. While their answers focus on Aspect #1 above, I'd like to focus on Aspect #2. 
Why do you want to normalize a constant to 1 in the first place?
As a physicist myself, I am familiar with doing this procedure and know that there are several reasons you may wish to do this. All properly defined unit systems are consistent with each other, so the decision to normalize any constant to 1 depends on what the unit system is designed for. There may be a pragmatic or a theoretical reason for doing this.  
Somebody working with spectra may wish to work in units where $h=1$, while a student of quantum mechanics may wish to define $\hbar = 1$. This is the pragmatic reason, wherein you just want to get rid of pesky constants. On the other hand, somebody trying to develop a unifying theory of everything may wish to develop a system of units where the "most fundamental" constant is set to 1. This is the theoretical reason, wherein you're seeking a "proper" or "elegant" representation, though extra factors may be introduced. 
