Our understanding of jet kinematics in high-redshift (z≥3) quasars is still rather limited, based on a sample of less than about fifty objects. It is difficult to perform such measurements for multiple reasons. In this work, we present very long baseline interferometry (VLBI) observations of the surprisingly rich radio jet structure of the powerful blazar J1429+5406 at z=3.015, observed at five different frequencies (1.7–15 GHz) between 1994 and 2018. While the outer jet components at ∼20–40 milliarcsecond (mas) angular separation from the core show no apparent proper motion, three components within 10 mas distance exhibit significant proper motions of (0.045–0.16) mas year^‑1, including one that is among the fastest-moving jet components at z≥3 known to date. The core brightness temperature well exceeds the equipartition limit, indicating Doppler-boosted radio emission. Based on the proper motion of the innermost component, we derive a small jet inclination with respect to the line of sight (within about 5°), confirming the blazar nature of the source. Recently, we analysed new high-sensitivity VLBI measurements of this blazar taken by the Very Long Baseline Array in 2024. At 1.5 GHz, the radio map clearly reveals a complex, extended structure around the source, reaching up to ~400 mas, rarely, if ever, seen in a high-redshift blazar. Here, we discuss possible explanations of this peculiar radio structure.