If the original poster is satisfied, that everything should be ok. However, I find this approach of giving a 'physical interpretation' of a purely mathematical idea slightly misleading. You can give a physical meaning to complex numbers, sure, but their mathematical meaning is far more interesting and compelling; I would rather speak of an application to physics.
As to fractional derivatives, they become quite easy to understand if you think that the Fourier transform takes the derivative of a function into multiplication by the variable: $widehat f'=ixicdot hat f$. So higher order derivatives can be defined as multiplication of $hat f$ by powers of $xi$, and it is no wonder that you can use this idea to define fractional derivatives, or actually generic 'functions of $d/dx$'. This leads to pseudodifferential operators etc.etc.
The main reason why this idea is not just a game but on the contrary is enormously useful, also in physics, is that using this kind of calculus you can give explicit (well, almost) expressions to fundamental things such as solutions to differential equations, and manipulate or estimate them in a very effective way.
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