Getting hilbert transform of IQ signal
Usually an IQ signal is generated from a real (RF, AF, etc.) signal by a radio (by quadrature heterodyne, Tayloe demodulation, complex vector arithmetic, or otherwise).  However, if you need IQ data, and the radio only generates I (or real) component samples, a Hilbert transform can be used to generate “fake” Q components (an approximation by making some assumptions about the original signal) to use in downstream IQ processing or analysis.
Thus, if you have an IQ signal, there is rarely any need to generate another fake similar Q signal component. Just use the Q component the radio provides instead of a Hilbert transform of the I component.  An IQ signal created by a Hilbert transform of the I component contains at most half the amount of information as contained by the original IQ signal (e.g. potentially aliasing the upper and lower sidebands).
And, to get a real signal back, just reverse whatever process (superhet, etc.) the radio used to generate IQ from RF.