The most general definition of a quantum state I found is (rephrasing the definition from Wikipedia)
Quantum states are represented by a ray in a finite- or infinite-dimensional Hilbert space over the complex numbers.
Moreover, we know that in order to have a useful representation we need to ensure that the vector representing the quantum state is a unit vector.
But in the definition above, they don't precise the norm (or the scalar product) associated with the Hilbert space considered. At first glance I though that the norm was not really important, but I realised yesterday that the norm was everywhere chosen to be the Euclidian norm (2-norm). Even the bra-ket notation seems to be made specifically for the euclidian norm.
My question: Why is the Euclidian norm used everywhere? Why not using an other norm? Does the Euclidian norm has useful properties that can be used in quantum mechanics that others don't?