Investigating the ion dynamics in the emerging bipolar pulse high power impulse magnetron sputtering (BP-HiPIMS) discharge is necessary and important for broadening its industrial applications. Recently, an optimized plasma source operating the BP-HiPIMS with an auxiliary anode and a solenoidal coil is proposed to enhance the plasma flux and energy, named as ACBP-HiPIMS (‘A’-anode, ‘C’-coil). In the present work, the temporal evolutions of the ion velocity distribution functions (IVDF) in BP-HiPIMS and ACBP-HiPIMS discharges are measured using a retarding field energy analyser (RFEA). For the BP-HiPIMS discharge, operated at various positive pulse voltages U+, the temporal evolutions of IVDFs illustrate that there are two high-energy peaks, E1 and E2, which are both lower than the applied U+. The ratio of the mean ion energy Ei,mean to the applied U+ is around 0.55-0.6 at various U+. In ACBP-HiPIMS discharge, the IVDF evolution shows three distinguishable stages which has the similar evolution trend with the floating potential Vf on the RFEA frontplate: (i) the stable stage with two high-energy peaks (E2 and E3 with energy respectively lower and higher than the applied U+ amplitude) when the floating potential Vf is close to the applied positive pulse voltage; (ii) the transition stage with low-energy populations when the Vf drops by ~20 V within ~10 μs; and (iii) the oscillation stage with alternating E2 and E3 populations and ever-present E1 population when the Vf slighly descreases unitl to the end of positive pulse. The comparison of IVDFs in BP-HiPIMS and ACBP-HiPIMS suggests that both the mean ion energy and high-energy ion flux have been effectively improved in ACBP-HiPIMS discharge. The formation of floating potential drop is explored using the Langmuir probe which may be attributed to the establishment of anode double layer structure.