If a unit dies with an equipment, and that equipment goes back to base, could I have reattach my equipment to another unit?

Yes, you can reattach that equipment to another unit on the same turn.

According to RiftJudge FAQ 9657:

"Yes, you can equip that equipment to another unit on the same turn after the original unit dies. When the unit dies, the equipment detaches and returns to your base in a ready state, allowing you to pay its equip cost to attach it to a new unit during your Main Phase when the chain is empty."

Sequence:

1. Unit Dies: When the unit dies, the attached equipment detaches and returns to your base.
2. State Check: Because it is no longer attached, its rules text is no longer inactive, and it becomes a ready card at your base.
3. Main Phase: During your Main Phase, provided the chain is empty (no showdown or other effects currently resolving), you can pay the equipment's Equip cost to attach it to a new unit you control.

As a reminder, you must perform this action during your Main Phase when the game state is Open (the chain is empty). You cannot do this during a showdown or while a chain is currently resolving.