Enemy probe teams may not attempt mind control here
Just so we're all clear here....Your faction can have ONE and only ONE headquarters at any given time. By default, your first base is also your faction headquarters. If you build a headquarters somewhere else, your original HQ goes away, "moving" to the new base. If your headquarters are destroyed or taken, you have to build a new one elsewhere. ( Which you'll want to do quickly, or face crippling inefficiency ). .
If things break your way, you will never need to build a headquarters. However, there are certain situations where it makes perfect sense to relocate headquarters to a new base. This is primarily useful if your base happened to end up on an island, peninsula, or uncomfortably close to another faction's border. In these cases, consider moving your headquarters to a more central location. This not only makes your headquarters better defended, but also can significantly reduce inefficiency at your more remote bases. ( Inefficiency is related to a base's distance from headquarters, so moving a significant number of bases closer to headquarters cuts inefficiency ).
Terminate with extreme prejudice any impulse to build headquarters at a vulnerable base to protect it against mind control. Far better to simply lose a base ( and take it back later ) than to deliberately put your headquarters in hostile cross hairs.