wavefunction - Quantum Wave Mechanics

I am studying QM-I these days. Now, I just think of the wave function as just a mathematical function that defines the state of the particle at an instant and from it you can extract various observables. I have been told that it is just a misnomer calling it an actual wave, and there is no wave there, its just that the wave function satisfies the general wave differential equation in 3d space. Now is this true? Is there really no wave? What about QFT, when we say electron is an excitation in a Quantum Field, do we mean to say, it is sort of a pulse in that field medium? Like a photon is a pulse in an EM wave and essentially electron is just a pulse, which we see as a particle? Or is there no wave there ? Is this related to $E=\gamma mc^2$ by relating energy in a wave to a particle.