We present high-resolution near-infrared imaging of the compact HII region N159-5 and its immediate environment in the giant-star forming region N159 in the LMC. N159-5 was observed at high spatial resolution ~0.11"-0.25" in the K-band using the ESO Very Large Telescope UT4 (VLT), equipped with the NAOS adaptive optics system. Our data reveal that N159-5 has a complex morphology formed mainly by two wings and probably a single central bright star, embedded in diffuse emission of ~4.5" diameter. A remarkable embedded tight cluster of approximatively the same size, containing at least 38 faint stars coinciding with N159-5, is also detected. Such clusters can be found in galactic HII regions like the star-forming regions SH2 269 or M42. At the location of the radio peak, especially in the bright western wing, this cluster is rich in stars. Spectroscopic observations reveal that the diffuse region is constituted mainly of dust continuum and that the bright star #2-55 could be of type O8 V. A comparison with the radio observation flux of N159-5 published in the literature seems to show that the bright star #2-55 is not the only ionization source of N159-5. Towards N159-5 molecular H_2_ emission is detected. A model of the region is proposed.