This question is [answered in comments](https://mathoverflow.net/questions/369154/is-simultaneous-similarity-of-matrices-independent-from-the-base-field?__=1674402206#comment931814_369154):

 > "As everyone is saying, this follows from Noether-Deuring. 
 > See [mathoverflow.net/questions/28469/hilbert-90-for-algebras](https://mathoverflow.net/questions/28469/hilbert-90-for-algebras) for a quick proof. 
 > I also asked this question a while back [math.stackexchange.com/questions/305696](https://math.stackexchange.com/questions/305696)." 

        – 
[David E Speyer](https://mathoverflow.net/users/297/david-e-speyer)

Thanks to all!