Yes, it is a very convenient representation, and it is possible to formulate quantum mechanics without using complex numbers at the fundamental level of the mat

Yes, it is a very convenient representation, and it is possible to formulate quantum mechanics without using complex numbers at the fundamental level of the mathematical representation. In standard quantum mechanics, the state of a quantum system is represented by a complex-valued wave function or a vector in a complex Hilbert space. Observables are represented by self-adjoint operators acting on this complex Hilbert space. The complex nature of the wave function gives rise to the phenomenon of quantum interference and the probabilistic interpretation of quantum mechanics. However, in this real quantum mechanics, the complex Hilbert space is now replaced by a real Hilbert space, and the complex wave function is replaced by a real-valued wave function or a vector in this real Hilbert space. The observables are represented by self-adjoint operators acting on the real Hilbert space. One way to achieve this is by using real algebra or real matrix representations. Now we let the complex numbers represented by 2x2 real matrices of the form: a + ib = [a -b] [b a] now a and b are real and the imaginary unit i is represented by the matrix: i = [0 -1] [1 0] One also can use quaternions, but now it will be more complicated without much to gain.