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 Quote by Edi I understand the system where the photon is replaced with .. something. A particle. System where the particle would simply have kinetic energy in another direction.. but can the same be for a photon? It would kinda have to move faster than c in order to cover that increased distance when summing the two vectors.
This is not correct. Let's simply work this out from the Lorentz Transforms.

If a pulse of light is moving at c in the +y' direction then the equation for its worldline is:
$$y'=ct'$$

If we substitute in the Lorentz transform expressions and simplify we get:
$$\frac{v}{c} x + \sqrt{1-\frac{v^2}{c^2}} y = c t$$

Noting that:
$$\left(\frac{v}{c}\right)^2 + \left(\sqrt{1-\frac{v^2}{c^2}} \right)^2 = 1$$
we immediately see that the speed is c in both frames.