It is a well-known theorem of Pettis (see, for instance, the following expository note of Andrew Putman https://www3.nd.edu/~andyp/notes/AutomaticContinuity.pdf) that any measurable homomorphism between Lie groups is automatically continuous (and hence, also smooth).
I would like to see some explicit constructions of non-measurable homomorphisms of, say, matrix Lie groups. For instance, how does one construct a non-measurable homomorphism $\phi : \mathrm{SL}(2, \mathbb{C}) \rightarrow \mathrm{SL}(2, \mathbb{C})$? I have heard that this can be done by applying some field automorphism (other than complex conjugation) to each of the entries, but I do not understand why this makes the homomorphism non-measurable (it is sufficient, by the above theorem of Pettis, to show that the homomorphism is not continuous).
Any details or references would be greatly appreciated!