"I moved two years ago" would be 二年前に引っ越した I think, but if I want to say after x time, would I use に？E.g. 四日後に, or is it not required?
In your examples, に is optional both after
～後【ご】. You can drop に and say 2年前引っ越した and 2年後引っ越すつもりです, although these sound relatively informal.
Note that when these words mean "before/after ～" rather than "～ ago/later", the particle choice may be more tricky. See: 後で vs. 前に- Why not the same particle? and What's the difference between に and で when speaking of time of an action?