Hilbert transform & the analytic signal
The Hilbert transform builds the analytic signal, a complex-valued signal whose real part is the original and whose imaginary part is the original phase-shifted by 90 degrees. From it one reads an unambiguous instantaneous amplitude (the modulus, that is, the envelope) and instantaneous phase (the argument) at every sample. This is the standard route to amplitude envelopes and phase time series for band power, phase-locking and phase-amplitude coupling analyses.
The construction is only meaningful for a narrowband signal: the analytic amplitude and phase have a clear physical interpretation only when the signal has first been band-pass filtered to a range narrow enough to have a single, well-defined instantaneous frequency (the Bedrosian and related conditions). Applied to broadband data it produces envelopes and phases that are mathematically defined but physiologically meaningless, and edge effects from the filtering must be discarded.
Hilbert-on-narrowband and complex-wavelet magnitude/phase give nearly equivalent results; the choice is one of convenience, not of principle. The interpretability failure on broadband signals is the single most common misuse.