In a Pathfinder campaign me and a friend are Co-DM'ing a campaign and we have a special fight planned for the party if they put in the extra work. They will fight 2, lvl 20, Mythic Tier III characters. At that Tier you get Amazing Initiative which you can spend a mythic point to give yourself an extra standard action for that turn. Now as a Weapon Master Fighter, using those two standard actions could I combo Unstoppable Strike and Deadly Stroke, and use them simultaneously for a single attack?
There is a way, but it doesn't rely on the class feature - your class won't matter.
Unstoppable Strike as a Weapon Master Fighter class feature allows you to spend a standard action to make an attack to allow that ignores many sources of AC. Because it is its own action, it can't be combined with another action.
Deadly Stroke is also a standard action to perform, and allows you to make a single attack against an impaired opponent to do double damage and bleed their Con. Normally, these two abilities cannot be combined.
However, Unstoppable Strike is not unique to the Weapon Master - while in normal games, very few classes get access to Unstoppable Strike, your characters are Mythic, and gain rare and powerful options from that designation. One of those options is the "Legendary Item" Universal Path Ability. This allows a Mythic character to bond to an magic item in such a way that it develops additional powers when wielded by a Mythic character.
Legendary Item can be taken twice at 3rd tier, advancing the item to a minor artifact. With that advancement, a variant of Unstoppable Strike can be added to it at that tier. The Legendary Item version of Unstoppable Strike requires you to spend the weapon's legendary power as part of an attack - not as a separate action. For a single legendary power, it works almost exactly like the Weapon Master's, just targeting touch AC. Against foes with magical deflection bonuses to their armor, though, you can spend two legendary power to also ignore deflection.
Because the Legendary version of Unstoppable Strike is an option when you make an attack instead of its own action, it can be combined with any attack you make, as long as you spend the legendary power.
It's not possible to use amazing initiative to combine a Deadly Stroke and an unstoppable strike
Although the extraordinary ability amazing initiative, in part, says that
as a free action on your turn, you can expend one use of mythic power to take an additional standard action during that turn.
...this special ability does not enable the creature to take simultaneously two standard actions, and both the the feat Deadly Stroke and the fighter archetype weapon master's extraordinary ability unstoppable strike mandate taking separate and distinct standard actions.
To be clear, just because a creature has two standard actions available does not mean the creature can combine two events that each take one standard action to perform into a lone event that takes a—let's call it—double standard action to perform. That option just isn't available through the special ability amazing initiative, and I'm pretty sure the game doesn't make such an option generally available at all.
However, a creature can use the special ability amazing initiative to make, on the same turn, a Deadly Stoke that's followed by an unstoppable strike. But I'm certain that's not what you want.
No you can't, according to the rules.
In a normal round, you can perform a standard action and a move action, or you can perform a full-round action. You can also perform one swift action and one or more free actions. You can always take a move action in place of a standard action.
In some situations (such as in a surprise round), you may be limited to taking only a single move action or standard action.
Amazing Initiative (Ex)
At 2nd tier, you gain a bonus on initiative checks equal to your mythic tier. In addition, as a free action on your turn, you can expend one use of mythic power to take an additional standard action during that turn. This additional standard action can't be used to cast a spell. You can't gain an extra action in this way more than once per round.
Deadly Stroke (combat)
Benefit: As a standard action, make a single attack with the weapon for which you have Greater Weapon Focus against a stunned or flat-footed opponent. If you hit, you deal double the normal damage and the target takes 1 point of Constitution bleed (see Conditions). The additional damage and bleed is not multiplied on a critical hit.
Unstoppable Strike (Ex)
At 19th level, a weapon master can take a standard action to make one attack with his chosen weapon as a touch attack that ignores damage reduction (or hardness, if attacking an object).
What you want to do is not possible, as both Deadly Combat and Unstopable force need you to spend a standard action. There is no rule allowing you to perform 2 standard actions as one standard action. It would be only possible if one of mentioned could be used as a free or swift action. Since it is not, you can't do what you want to do.