The most basic fundamental concepts of  quantum physics are presented by means of a simple and pedagogical example, namely the Euclidean plane viewed as  the Hilbert space of quantum states. 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. Because the set of pure states or a set of mixed states  solve the identity, they are used  to quantize functions on the unit circle and to give a semi-classical portrait of quantum observables. Since the tensor product of two planes, their direct sum,  their cartesian product, are isomorphic (2 is the unique solution to $x^x= x X x = x+x$), and they are also isomorphic to $C^2$, and to the quaternion field $H$ (as a vector space), I will describe an interesting relation between  entanglement of real states, one-half spin cat states, and unit-norm quaternions which form  the group $SU(2)$. I will explain the most general form of the Hamiltonian in the real plane by considering the quantization of a magnetic-like interaction potential viewed as a classical observable on the unit 2-sphere. Finally, I will present an example of quantum measurement with  pointer states lying also in the Euclidean plane.

