We numerically investigate the pulsating dynamics of pure-quartic solitons (PQSs) in a passively mode-locked fiber laser. The bifurcation diagrams show that the PQS can alternate between the stable single soliton and pulsating regimes multiple times before transiting into the chaotic state. This multi-alternation behavior can be attributed to energy redistribution across the central part and the oscillating tails of the PQS, which is caused by an imperfect counterbalance between self-phase modulation (SPM)-induced and fourth-order dispersion (FOD)-induced phase shifts. Soliton creeping behavior can be observed during the pulsating process, accompanied by periodic asymmetric temporal profiles and central wavelength shifts of the PQS. These findings give new insights into the dynamics of PQSs in fiber lasers.