Mega Man X Legacy Collection

So although the game only lets you get one armor enhancement capsule, it is possible to get 3 of the 4 via a Pass Word which is almost as good as having the Hyper Chip. The only armor enhancement the following code does not have is the arm upgrade as I felt the helmet, armor, and leg upgrades are more useful. Additionally, this code also has the Z-Saber.
5114
6685
6872
3115

I recommend entering the Sigma fight with the Z-Saber ready to go. Dodge his first attack of fireballs, then when he jumps up to the center of the room and fires down, hit him with the saber. Then charge up to green and get rid of the first 2 shots and hit him again just like the first time to finish him off.

The second form is a bit tougher because his head is so small and elevated and the mines he fires do a ton of damage. It is possible to hit him from either side but it’s easier from the left side so his back isn’t protecting his head. At the start of the fight, charge up to green and ditch the first 2 shots again as he moves to the left side. Dodge his mines and missiles and once he moves back to the right, jump over from the top left wall and saber him in the face. Do that twice to finish him off for the achievement. Of course, since this is the last battle, use any sub tanks as needed.

EDIT: Beating Kaiser Sigma on Rookie Hunter Mode without the Hyper Chip WORKS! He still damages you in the fight but if you stock up on energy for the Sub-Tanks they'll keep you alive throughout the fight.

BIG thanks to DarkWingSpartan and HoarderKhaos for the advice, you two really had a great idea!

Original Post (for those who don't want to use RHM): WORST! SIGMA! INCARNATION! EVER! (If you don't count X7's Sigma battle). And you cannot, I repeat, CANNOT have the Hyper Chip on you. You definitely want all four Sub-tanks full for this, it's going to be a pain. Also, I wouldn't have Zero's Z-saber on you (it doesn't help, really, unless you are highly sufficient with it).

First form - Sigma's got a shield to block attacks, while he attacks with fire that runs up along the wall across from him and each time he shoots he brings the shield back up for a second to block before he does it again (a total of 6-7 times) before he jumps across the room, pausing in the center only to fire three more times AND depending on where you're standing it'll be to the left, the right, or the center straight down (and if he does that, his fire splits and runs along teh floor and walls after you).

Soon as the fight starts climb up the wall behind you to the top, when the first shot nears you, jump off and wait a sec or two before you air dash upward and then move right to avoid the last shot, and then quickly dash back to your side of the room before Sigma jumps, aim away form him, and fire the Spinning Blade to he jumps right into the upper of the two. If you're lucky, he'll aim his gun towards your side and miss you completely; otherwise he'll aim straight down and that's when you jump up and air dash upward to avoid teh shots. Then quickly scale the wall away from Sigma. Repeat the process.

After half his health goes down, Sigma starts tossing his shield before he fires; after he's down to 1/4 his health, the shield will fly in a figure 8 pattern once before returning to Sigma. As longa s you wall-climb up away, you'll avoid it. Try not to use any Sub-tanks here (or at least use only 1)

Second Form - Special weapons can't help you at all, because Sigma goes into a HUGE battle body with his small head beating the only weak spot... and the only way you can hit him is with Charged X-Buster shots. The trick is to aim squarely at his massive shoulders so your shots hit his head (which, oddly, is a little below the shoulders, go figure).

Sigma will NOT make it easy for you - he flies from one half of the room to the next, shooting explosive mines at you. When you're behind him, he fires missiles either four at once, or one at a time for a total of four, and they deal a lot of damage (5 or more bars of your health). Eitehr dodge or blow them away. When he's in front of you, he'll fire a wide beam either downwards or upwards (depends on where you are, the floor or on the wall) that will eat your health like a cop on donuts.

I recommend not fully charging your X-Buster, just regularly charged ones like before you got the Arm Upgrade. Normal shots (no charging required) will just bounce off Sigma. Watch your health, corss your fingers, and pray if you got 'em. I lost count how many times Sigma killed me (sadly, I had the hyper chip on me so I have to go back and replay the game again).

Good hunting, and good luck!

I've found that you can still use OTHER chips, just not the Hyper chip when beating Kaiser Sigma.

This strategy requires the Leg Chip (for double aerial dash).

When fighting his first phase (with Z Saber): Climb up the wall, then double dash in the air to the opposite one (so you're now above Sigma). As he fires the sixth flame, jump over him and land behind it (so it's heading up the wall as you land on the ground). Now, wait until Sigma jumps to launch 3 more flames. While he's hovering, hit his ankle with the Saber. Repeat twice to kill him and move onto Kaiser Sigma.

When fighting his first phase without the Z Saber, stay on the wall until he jumps. then fire charged shots into his back, and if you're not on the ground he won't fire followup flames. After he jumps, double air dash to the wall behind him and start charging up again.

When done correctly, Sigma isn't able to touch you for the entire first phase.

Kaiser Sigma

This is actually much more simple than it appears (despite the difficulty in hitting him).

Dash under his feet when he hovers to the left, and keep a charged shot ready for the missile barrage he will fire after he lands. Keep an eye on the mines he's launching as some will "bounce" once and go for you. The ray he launches is very easy to dodge, just try to have him aim it at the ground so you can keep shooting at him while using air dashes as needed to avoid mines. If you'd rather take it slower and safer, have him aim upwards and you won't deal with as many mines because the ray will hit them as well.

If trying to use the Z Saber to hit him, I've found it's easiest to time when he starts hovering to the left and then swinging it mid air so that the slash itself hits his head. Otherwise, just blast him normally.