Is Turn Watcher 1.4 compatible with D&D 4e?

Yes. Also by default we show 3.5e columns, you can now edit these columns to your taste and make them compatible with different versions of d20 such as 4e and 5e of Dungeons & Dragons.

For additional information you may want to read the Initiative pages that describe all the details about initiative of all the versions of D&D from the very first version in 1974 up to 5e at time of writing.