The Doctor Who Xmas special came and went and... well... after a number of fairly uninspired ones, as in: I watch them once then never have to spend time looking at them again, we were offered "Doctor Who meets Superman".
While I won't go into anything spoiler-ific the previous statement has not yet told you.
The basic plot could have been interesting, if the Doctor OR Superman had been left to handle it by themselves. Instead, we got a half-and-half mix of both, an awkward situation lasting way too long. Not unlike when someone farts in an elevator and there is only one other person. Except the smell lingers and lingers and won't go away.
This is typical of the Moffat-era of Doctor Who: we are given to interesting plots that un-mistakenly end up with a "meh" of indifference.
The cast of characters
(to show how much I didn't pay attention, here are the name I know them by):
"Superman": A kids who swallows a gem. What kind of stupid, dumbass eight-year old kid just swallows a quarter-sized gem? Then the guy becomes a (further) complete(r) loser by become... a nanny. Talk about a creepy job! A male nanny is something I simply would not consider for my kids, ever. In the history of mankind. E-V-E-R. NOPE. Superman is a nanny! At that point I was pretty much done with it. The worse about it? Knowing Moffat, we will be treated to a return of this guy. And it won't be a cool welcome, like when Russel T Davies brought back Rose, Sarah Jane, or Torchwood back to fight the daleks, and all contribute to the plot in a realistic and significant ways.
"Lois Lane": The lady did a great job keeping a straight face with the stupid crap that happened around her. She had to drink a lot to become that jaded, proving that she is a good actress.
"The Fat bald dude": Now wtf was he doing there? He just added to the general confusion of the situation, not really adding much and robbing us of a scene where the Doctor opened up about his feelings about River Song. A waste of space who knew things he never should've known according to HIS timeline. After all, he ended up in a robot at the end of "The Husbands of River Song".
"The Villains": These guys had decent plan, one that would work well in Torchwood. These guys had no chance ever facing Superman or the Doctor, much less the two by themselves: a wasted "Body Snatchers" opportunity for a weak plot hook.
The Doctor: Peter Capaldi does what he can with such terrible material.
How I would fix it?
I really, really, spent time wondering whether it was worth it or not. It's not really worth it. Yet, since this post is call "How I'd fix it", I do have to make an effort.
Here is what I would've done:
The intro scene remains effectively the same: the Doctor making his thingamajig, dumbass kid swallows gem, Doctor freaks out and tells kid to wait for him here.
Then we see the kid as a man, hopefully with a less loser job than nanny: perhaps he's a handiman of her building or a concierge. Not overly glamorous, but something that gives him access to the girl, but from afar. I could even see him use his powers to make sure she needs to keep calling him: clogged sinks, scolding water, freezing water, etc. Since she is busy with her job, she gets him to do a bunch of things for her. The rest of the people complain about the crummy service of the building.
We quickly catch up with him as Superman: saving kittens, stopping fires, etc. Lois sees him as a bad guy who is a renegade. She's not in love with him, she believes he's an alien who is part of an alien invasion (this would nod to "Love and Monsters", maybe she is the sister/cousin/friend of someone therein). She drives around town trying to find Superman to see where he has his "lair".
She has Clark install a series of cameras and other sensor on the roof to detect where he comes and goes. She tells him how much he hates aliens and wants to save mankind from them (this is a Xmas special so she must be "redeemable").
At her job, she is sent to meet with the new local scientific wonder: pretty much like the episode. We get the official story of how the company will help the world by [blah blah blah]. While there, she sees something odd and hides herself in one of the many conveniently-located closets and waits for everyone to leave. Once everyone leaves, she witnesses the scene with the two men where one becomes a brain implant.
Then, Superman shows up and speaks to them (but she can't hear what they are saying). Superman is responding to the pre-brain change distress call. They have a quick chat but then seeing nothing is wrong, Superman leaves. The woman is now convinced that Superman is part of the alien plot.
She runs home to tell Clark everything she found, but he is not there, so she sits at her computer desk looking for Superman. The cameras and sensors capture the Tardis appearing on the roof and the Doctor looking around. She watches as the Doctor spots the camera, approaches it. After a moment, he points the sonic screwdriver at the lense and asks "Are you watching this? Of course you are! I have returned! Come get me I'm here on the roof! Come on my little Superman!"
The woman is pretty freaked so she walks around for a moment, going crazy, biting her nails looking for something she does not find (assume a gun). The Doctor keeps teasing her to come upstairs.
* * *
We go back in time to the Doctor stepping into the Tardis at the start of the intro scene. He plays with the Tardis' dials and ends up on a planet where he goes to meet with some "elves" (remember, this is a Xmas special) who build things. Walking around a fantastical super sci-fi workshop, he meets "the Boss" who is Santa Claus, but we only see him from being and never fully. There is a big screen with a list of name and the word "naughty" at the top of it.
The Doctor recaps the situation. On a computer, he adds the Doctor's name to a "naughty list" that makes him target for bounty hunters of all types to hunt him. He tells the Doctor must give to get what he wants. The Doctor rants and raves about it. Then snaps his fingers and tells "The Boss" to wait and he runs back to the Tardis.
* * *
We come back to the present. The Doctor sees Superman flying into the building. "Ah! So he's done it." He turns to the security camera again. "Come! Come! I know you are there."
The door opens and out comes Lois. Seeing her, the Doctor goes "See, you got yourself a better disguise! No one knows your Clark Kent is Superman!" The Doctor talks to Lois as if she were Superman, effectively telling her Superman's story. Lois says very little. As the Doctor lets her know that he is an alien and that Superman is a human who was changed by an alien and not part of an alien conspiracy.
The aliens show up. "You thought you would escape us, Miss Lane. Fortunately we have security systems that showed us your visit." Using alien guns, they threaten to turn the Doctor and Lois into alien bodies. The Doctor full of bravado jests with the alien that he is no longer the sole protector of the Earth. During that time, he keeps winking at Lois as if saying "change into Superman". Opening their head, the aliens divulge their plan to drop a spaceship on NYC and that there is nothing they can do.
Aliens stand around as a bright star is seen in the sky. The aliens say [muhahaha megalomaniac rant about taking over the world]. As the aliens look up, the Doctor jumps into the Tardis. Lois follows without the Doctor seeing her.
One of the alien brute calls to the leader. "We may have a problem, you need to come see this." And the aliens run out of the roof, leaving the tenants looking up and pointing to the sky. Some might make a comment about how "These people in London are really sparing to expenses"
* * *
The leader looks back down not seeing anyone. "You cannot stop us now, our plan will succeed muhahaha soon Earth will belong to us!" Into a communication device, he says "bring anyone you find to me, there is no way they could've gone anywhere. They may be hiding." The alien brutes appear on the roof, pushing the tenants. "Where is the old man and the woman," he asks without answer. The tenants have no idea what they are talking about.
The Tardis reappears on the falling ship. As he gets to the cockpit, he begins to play with the controls, hoping to stop the descent. "I thought this would fly like the Titanic." Then we can see things are not going well. Lois enters the cockpit, shocked at the sight before her. The Doctor keeps talking to himself talking about how "Alas poor Adric, now I understand this final flight of yours." He asks her to do a few things "press that, pull this, will you?" before realizing that she is there. "Shouldn't you be out there stopping this rock from flattening New York?"
Seeing her blank face, he understands his mistake.
In the background, the Earth getting bigger and bigger with the ship obviously going to New York. The Doctor freaks out.
* * *
The aliens crowd into Lois' apartment before the screens an alien is at the keyboard. "All the data recorded on here has many backups. And you told our entire plan to the world!" The aliens begin arguing. The camera showing the roof is getting brighter and brighter. "Let's get out of here!" the leader says and they all teleport away (the device could resemble Jack Harkness's teleport gauntlet).
The screen itself get blindingly white.
* * *
The Doctor is sitting at the controls, Lois grasps the chair. In the front windows, the roof with the tenants stand cowering, slowly they begin pointing to the ship and cheering. The Doctor straightens himself. "That's what I'd call a Christmas miracle!"
Lois looks outside. "Yes, indeed. A Xmas miracle!" The Doctor notices that she is not looking at him.
The scene changes to Clark Kent holding up the ship flying just above the cheering tenants.
* * *
Later, Superman comes back to the roof, having disposed of the ship. Lois jumps into his arms. "I was so wrong. You are not an alien menace, you are the protector of mankind." Superman laughs. "Oh! Yes! I came for... Oh yeah! I need to give you something of value..." Superman kisses Lois. "Oh! I guess there is no point now." The Doctor reminds them to be careful. Lois tells the Doctor that they will spend their time hunting these aliens to make sure they don't cause any further problems.
The Doctor jumps back into the Tardis.
* * *
In the Boss's office, the Doctor says that he doesn't need the crystal anymore, that Earth already has one. The Doctor then leaves the office. On a screen before the Boss, he deletes "The Doctor" from the naughty list and gives a big Ho ho ho.
Not great? I did say I would NOT have the Doctor meet Superman... But I think the story I presented here would have been significantly better and more interesting.