ΔE 17.9, loose: different enough that you should pick on purpose. Amethyst Purple is a little lighter, much more saturated, warmer than Emperor's Purple.
Not as a straight swap; see the closer alternatives below. Loose match (ΔE 18). Amethyst Purple is a little lighter, much more saturated, warmer than Emperor's Purple.
The same difference applies the other way round: Emperor's Purple is a little darker, much more muted, cooler.
ΔE is the colour difference between our two swatch estimates: under about 6 is close, over about 15 is loose. Real paints also differ in finish, coverage and how they thin, so test before you commit.
See Emperor's Purple on the colour wheel See Amethyst Purple on the colour wheel
All matches: Emperor's Purple equivalents · Amethyst Purple equivalents