All my life I’ve seen the word into used in this way for written communication:
I’m really into this.
Are you going into work today?
For me, into just doesn’t seem right in those sentences. I think in to would be more correct,
I’m really in to this.
Are you going in to work today?
In the above two examples, is into or in to correct?
Both are idiomatic expressions. Neither of them are to be taken literally. If you are going into work, you are not outside the building about to enter through the doorway. If you are into jazz, you are not physically moving through a jazz dimension. As idioms they can be spelled any way the historic use of the term settled on, so into is fine.
There is even a dictionary definition that fits, as per dictionary.com:
(used to indicate entry, inclusion, or introduction in a place or condition): received into the church.
Since we’re citing the dictionary, Merriam-Webster gives one definition of “into” as
So I would write “I’m really into this” (one word).
But I would say “Are you going in to work today?” because “in” and “to” do function as separate words:
“Are you going in today?”
“In where?”
“To work.”
I think those should both be “into” since they are positioning the object, even if just idiomatically. The second one might be “in to” if work is a verb and not a place.
That would be into.
“To do what?”
“To work.”
would be “in to”.
You didn’t smoke enough reefer, Daddy-o. 
You are if Rod Serling says it.
Here’s the way I see it–
“Into” is a colloquial term for “liking something.”
“In to” refers to physical movement.
Gallant: “He went in to escape the rain.”
Goofus: “He went into escape the rain.”
So does “into”
“In to” means to go in somewhere to do an action (verb)
“Into” means to go inside a place or thing (noun)
I think the first one in the OP is right.
I think the second should be in to.
The reason is that I think “going in” is the idiom. He went in, she’s going in. To work, to school tomorrow. They are separate phrases to me.
@Thudlow_Boink is correct: “I’m going in to work today.” It may feel like you should use “into” here (based mostly on how easily the syllables roll together), but that’s incorrect. The words should be separate and discrete in this usage.
If you take ‘going in’ as a phrasal verb, then ‘going in to work’ is correct.
You could equally well say,
I’m going to work
I’m going in to work
‘Going in’ is a verb on it’s own, even though it’s two words.
However, in sentences like ‘I’m going into the bedroom’, ‘into’ is correct because it indicates entering a place.
In the one case ‘in’ is part of a phrasal verb, in the other ‘into’ is a preposition.
By the time I read the first half-dozen replies, I realised that I had no idea which is correct. As with so much grammar, I just use what looks right to me, fully appreciating that other’s (including grammar correction apps) may disagree.
If “work” is used as a place then it would also be “I’m going into work.” just as you would say “I’m going into school.”
There’s a difference.
I’m going in to work = I’m travelling to my workplace
I’m going into work = I’m now entering work, or my workplace
I’m going in to school = I’m travelling to school
I’m going into school = I’m now entering the school grounds
In the first case the verb is ‘going in’ and the preposition is ‘to’.
In the second case the verb is ‘going’ and the preposition is ‘into’.
Why the “now”?
Some people have to work on Saturdays.