  We report the discovery of three double-peaked Lyman-$\alpha$ emitters (LAEs)
exhibiting strong blue peak emission at 2.9 $\lesssim z \lesssim$ 4.8, in the
VLT/MUSE data obtained as part of the Middle Ages Galaxy Properties with
Integral Field Spectroscopy (MAGPI) survey. These strong blue peak systems
provide a unique window into the scattering of Lyman-$\alpha$ photons by
neutral hydrogen (HI), suggesting gas inflows along the line-of-sight and low
HI column density. Two of them at $z=2.9$ and $z=3.6$ are spatially extended
halos with their core regions clearly exhibiting stronger blue peak emissions
than the red peak. However, spatial variations in the peak ratio and peak
separation are evident over $25\times 26$ kpc ($z=2.9$) and $19\times28$ kpc
($z=3.6$) regions in these extended halos. Notably, these systems do not fall
in the regime of Lyman-$\alpha$ blobs or nebulae. To the best of our knowledge,
such a Lyman-$\alpha$ halo with a dominant blue core has not been observed
previously. In contrast, the LAE at $z\sim4.8$ is a compact system spanning a
$9\times9$ kpc region and stands as the highest-redshift strong blue peak
emitter ever detected. The peak separation of the bright cores in these three
systems ranges from $\Delta_{\mathrm{peak}}\sim370$ to $660$ km/s. The observed
overall trend of decreasing peak separation with increasing radius is supposed
to be controlled by HI column density and gas covering fraction. Based on
various estimations, in contrast to the compact LAE, our halos are found to be
good candidates for LyC leakers. These findings shed light on the complex
interplay between Lyman-$\alpha$ emission, gas kinematics, and ionising
radiation properties, offering valuable insights into the evolution and nature
of high-redshift galaxies.
