Analyze and Experiment.

You spent some time writing down analysis of how Alex felt your powers had affected her. Your 'mental twin' was quite thorough with her description, the way she explained it, it was like someone pulled apart 'her' and 'him' then stitched them back together, her first idea was like a quilt.

But after thinking for a moment, she came with a bit more clear concept, though it was more confusing. "It's like those illusion patterns, where if you look at it one way, it's 'Alex', but with a slight shift, you see 'Arthur'. But the way it's mixed, it's like 2 structures which reinforces each other, they cover the issues the other has." She said.

"Alex feels confident about her body, but she's also a bit tense, not letting herself relax, while Arthur don't feel confident, but he 100% knows how to calm down and relax, in order to focus. Combined, I have the Confidence of Alex, but the calmer demeanor of you. And if I switch to you, I have the openness to flirt, while also knowing how a guy would feel about the flirting." She finished

Basically, the 'combined' personality is stronger then either one alone. She was probably stronger then you, both physically and mentally. That brought up a thought, why was she listening to you?

"So you'd argue you are mentally stronger then me, right?" you asked, deciding on a test. "Like you wouldn't do anything you don't want?"

"Ya, probably." She said with a confident smirk. This acted like a challenge to you, so you decided to go forward with the test.

"Alright then, Take off your shirt." You said, a minor tone change you didn't know the origin of. It was like your voice took on authority, and Alex seemed to think so too.

She seemed to shiver at the order, and then pulled off her shirt in a rush. Alex was wearing a gray sports bra underneath, she had a slightly bewildered look, before it clicked and she grinned. Seems like she had no issue with you being in control.

"Oh I get it, you wanted to test if you were in control of me?" Alex said, figuring out your major question. "Well I guess you can make me feel what you want, it's like I'm a cloned device, whatever you do/want, is reflected on me."

"So I guess that solves that issue. Let's try writing down some ideas, but it's also getting a bit late. Could you like to get some food for us, while I plan out some tests?" You said, trusting her to know what food you two would like. She seemed to think before grabbing her keys and beginning for the door.

"Ahem" you interrupted, gesturing for the shirt she left. She rolled her eyes before grabbing it and putting it back on. Then went to grab some food for the both of you.

Ms. Moore return not 25 minutes later, it was a bit early for lunch, she bought sandwiches from a local store, bringing some custom orders for you both. The place wasn't that far, but it allowed you to properly test your powers, to which you cleared out that there was definitely a range to your powers. You also wrote down some ideas, as you said.
"Alright first up, let's see what exactly can be 'swapped', swapping the mind can't be the only thing, can it?" You said, having prepped for your tests.

"Gotta ya, what shall we try?" She said, putting the sandwiches on the countertop by the stove. You held out your right hand and she grabbed it in return.

"Just unlock my phone with your fingerprint." you said, having tried your first thing, and feeling a strange sensation for a moment as you tried your powers in a new way.

"What do you..." Alex said, looking at her right hand, before realizing the change. "You gave me your hand?!" she said in a whispered shock. She immediately tried your phone and unlocked it. She was completely bewildered at your power. Then as she looked at her hand, the size being off for her body, like she was wearing a giant glove.

You grabbed the hand again and her hand returned to normal, no longer having your hand. "Now try to give my her body." you said, asking to see if she maintained your powers.

She nodded before trying to focus, you felt a tingling sensation across your body, your viewpoint shifted as your height lowered. Your clothes hung loosely to your body as you felt her female form take shape. You looked down in mild shock, surprised that not only it worked, but you now had 2 copies of Alex in your apartment.

You stepped into the bathroom, looking at the mirror, you saw Alex in guy's clothes. You tried speaking and it seemed you even got her voice. Your body felt small, the clothing felt weird of her skin, the sensations were overwhelming you.

You decided to try and 'remove' her body, to which it was rather easy, like hitting the 'undo' button. Your body returned to normal, but you had to readjust your clothes, they weren't sitting right after undoing the transformation.

Alex however, had read your other ideas and tried something while you were busy.

"Uh, You need to see this."

You walked over and looked at the phone in her hand, you had written down the question 'could you use your powers on technology?' for another test. The answer was yes, but not in the same way as your 'projections' worked.

It looked like an app opened on her phone, almost like a character creator, showing Alex's body in an idle position. A bunch of different options were appearing along the border, you pulled up the different tabs, but you didn't feel connected to the device like you were to Alex.

Among the options you saw there was stuff for Appearance(clothing, height, hair styling, etc),Build(stuff like muscle strength vs muscle definition), Sexual preferences(anything from how romantic she wants things to be, to kinks and turnoffs.), Mind(Text editor for memories, present thoughts, and more). As well as Grayed out 'undo' and 'return to default' buttons.
There was also a section for what style of changes, from "new haircut" where people think it's a mild difference, "Retroactive" which would make everyone perceive things as if nothing changed but adjust to the changes, and even "none" where people would notice any major changes immediately.

However, you couldn't seem to get out of the program to the rest of the phone. It was like Alex overrode the programming of the phone to use this. You tried injecting your power into the device, but that just unlocked your model and did nothing else. An option did unlock though, it seemed to save both models as 'defaults' to the program.

You tried to 'undo' the change to the device, and it returned to normal. You set down Alex's phone, and picked up your own, then tried to add the app to your phone. It looked the same as on Alex's phone, even having both models ready for use. After returning your phone to normal, you wrote down some notes for later testing. Going down this rabbit hole could wait.

You looked at the next experiment: "See what happens to Alex when you pull out your mental copy" with an additional note *Prepare proof you didn't do anything to Alex in case she gets mad!*