Imagine that we have an object with apparent magnitude $m_1$, later, we observe that the same object appears with an apparent magnitude $m_2 = m_1 + \alpha$, then what we can say about the absolute magnitude M? By the definition I believe we find that the absolute magnitude after the change in apparent magnitude will be $M_2 = M_1 + \alpha$. But this make sense? I mean, the absolute magnitude is that, absolute, what are your thoughts in this? What will be the change in the absolute magnitude after a change in the apparent magnitude?
The absolute and apparent magnitude are simply related by the distance to the source.
If the distance is fixed and the apparent magnitude changes, then the absolute magnitude must have changed by the same amount.
The absolute magnitude is not called that because it can never change - there are lots of examples of variable stars. It merely refers to the fact that it is the apparent magnitude if the object were at a defined distance of 10 pc.