I wonder how this question passed unnoticed and uncorrected...
Nope you actually cannot. Talking for Hungarian, German, Romanian and Polish, German seems to be the closest to get it working the way you described. Indeed, there is bármi and akármi but you can't use them with this meaning. "Te látsz akármit/bármit?" would mean "Do you see anything you want/anything anyone can ask for?". "Te látsz bármit is?", with appropriate intonation, can sound much closer to what we are aiming for, like "Do you see anything at all??". Still, rather uncommon. It's good to know that it exists but I wouldn't call it the way to go choice.
In languages I know, equivalents of "anything" can only be used in the "I can do anything" meaning - the "think of something - I've got it covered" meaning. Not the "fill the gap" meaning as in "Can you see anything (it's not my task to know what exactly, fill the gap for yourself)".
In this example, with "Do you see __?" there's not much difference between the two, but in a lot of contexts, there is a huge difference between "some" (indefinite but specific) and "any" (indefinite and non-specific).
Think of the sentences "Do you have something for me?" and "Do you have anything for me?" In the first, the speaker has something specific in mind. In the second, the speaker doesn't and it doesn't matter what the identity of the possible thing is.
I'm not an authority on Hungarian in any way, as I'm only up to here in the course and new almost nothing about it before I started, but from Wiktionary:
valami = something
akármi / bármi = anything, whatever, (whatsoever)
... it looks like Hungarian also has a distinction in these words. This may not be the best sentence to teach it with, because we use them pretty much interchangeably in this context, but still, I think there's a reason for not accepting "anything" in this sentence.
I think in English there is a difference. If I say "Do you see something?" I am hopeful the person sees something. I THINK they might be seeing something.
If I say "Do you see anything?" I am hoping that the person sees something. But I am doubting that they will see anything.