  The aim of the present paper is to introduce and to discuss the most basic
fundamental concepts of quantum physics by means of a simple and pedagogical
example. An appreciable part of its content presents original results. We start
with the Euclidean plane which is certainly a paradigmatic example of a Hilbert
space. The pure states form the unit circle (actually a half of it), the mixed
states form the unit disk (actually a half of it), and rotations in the plane
rule time evolution through Majorana-like equations involving only real
quantities for closed and open systems. The set of pure states or a set of
mixed states solve the identity and they are used for understanding the concept
of integral quantization of functions on the unit circle and to give a
semi-classical portrait of quantum observables. Interesting probabilistic
aspects are developed. Since the tensor product of two planes, their direct
sum, their cartesian product, are isomorphic (2 is the unique solution to x^x=
x\times x = x+x), and they are also isomorphic to C^2, and to the quaternion
field H (as a vector space), we describe an interesting relation between
entanglement of real states, one-half spin cat states, and unit-norm
quaternions which form the group SU$(2)$. We explain the most general form of
the Hamiltonian in the real plane by considering the integral quantization of a
magnetic-like interaction potential viewed as a classical observable on the
unit $2$-sphere. Finally, we present an example of quantum measurement with
pointer states lying also in the Euclidean plane.
