"Red oxygen" is the nickname for O$_8$. It only exists at extremely high pressures.
What kind of bond does it have? Can hydrogen theoretically have something similar at very high pressures too?
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That is what I found on wikipedia. It has a cubic structure and it was not predicted theoretically. here the pressure is not very high but is high(10 GPa)
High-pressure studies of hydrogen typically employ a diamond anvil cell (DAC) made of two diamonds holding a few-micrometer-thick gasket containing a tiny sample of hydrogen.
These are some more images I could found of hydrogen at high pressures, but I do not think any of them are like red oxygen.
Hydrogen forms similar clusters as cations (here). Some do not even need high pressure. The trihydrogen cation, with a triangular structure resembling an aromatic cyclopropenyl cation ring, is found in interstellar space and some planetary atmospheres, and the six-atom cation with its peculiar "turnstile" strcture (two triangles joined vertex to vertex and twisted 90 degrees from each other) is observed in solid hydrogen.