Is it possible to get the $\alpha$ and $\beta$ (amplitudes) of a qubit in Qiskit, i.e. not only probabilities after measurement but complete quantum state?
This can be done using the
statevector_simulator provided with Qiskit Aer. It will return the statevector that describes the quantum state at the end of your circuit. It can be used in the same way as the
qasm_simulator, only your circuit shouldn't have measurement gates at the end. There is more information about this simulator in this tutorial.