I think it is believed that we can make a time machine if we move faster than the speed of light. Obviously we know light travels at lesser speeds in mediums of high refractive indices. Is that correct or I have misread the concept?

I think it is believed that we can make a time machine if we move faster than the speed of light. Obviously we know light travels at lesser speeds in mediums of high refractive indices. Is that correct or I have misread the concept?
It you naively insert a superluminal speed into the equation for "time dilation" in special relativity you get an imaginary time dilation, not a negative time dilation.
Photons always travel at c, even in a medium.
While the phase velocity of light through a medium may be less than c that doesn't help you anyway, as the "speed of light" as the term is used in relativity refers to the speed of light in a vacuum.
While there appears to be nothing in the general theory of relativity that prohibits closed timelike curves (the criteria for "time travel"), it is generally believed, though not proved, that such are not possible. The impossibility of closed timelike curves is the "chronology protection conjecture" of Steven Hawking  which is the conjecture that macroscopic closed timelike curves are impossible. Hawking has proved this conjecture under the "weak energy condition", which seems to be reasonable at macroscopic scales, though not at the scales of quantum physics.
Note that there are exotic solutions of the Einstein field equations that admit closed timelike curves, for instance behind the event horizon of Kerr black holes. So more than general relativity must be involved in any proof of the chronology protection conjecture.
Closed timelike curve  Wikipedia, the free encyclopedia
Phys. Rev. D 46, 603 (1992): Chronology protection conjecture
Is time travel allowed?  plus.maths.org
[0801.0735] Topological censorship and chronology protection
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