Formal:
* Regardless of
* Notwithstanding
* In spite of
* Despite
* Irrespective of
* In any event
* Come what may
Informal:
* No ifs, ands, or buts
* By all means
* Under any circumstances
* Through thick and thin
* Rain or shine
* Whatever happens
Figurative:
* Through fire and water
* Hell or high water
* Til the cows come home
* Until the end of time
Specific situations:
* For a decision: I'm doing this, no matter what anyone says.
* For a promise: I will always be there for you, no matter what.
* For a challenge: I'm going to finish this project, no matter how long it takes.
Choose the phrase that best fits the situation and your desired tone.