Is it ok in Danish to have two verbs with different prepositions and a shared object like in this sentence without repeating the object?
The English sentence sounds strange to me. Admittedly, it might be just my fault, but there's a rule in my first language (Czech), that you need to repeat the object if the verbs use a different preposition (and in the case of Czech possibly a different grammatical case as well), eg. something along the lines of I will neither see it nor think about it. You can skip the repetition only if the prepositions (and cases) would be identical.
My question is: am I blindly projecting the rules of one language to others, or are both versions acceptable in English and/or Danish, or is it just a matter of personal preference and style, or am I onto something?
Well, I'm not so sure about the rules in English, but I think you are right that you would usually put in both preposition and object.
This is not the case in Danish, however. If you have two verb+preposition+object constructs like in the sentence above, you can skip the object and also the preposition if they are the same (they are in that sentence: se på + tænke på). If the prepositions are different you can choose to just state the two prepositions and not the object twice.
For instance: 'Jeg vil hverken tale om eller tænke på det'. In spoken Danish, I think it is more common to hear both prepositions and objects: 'Jeg vil hverken tale om det eller tænke på det'
Thank you! That's as I thought. Since the form of the object is the same regardless of the preposition, there's no need to repeat it.
What you wrote made me go through the entry about the verb on Ordnet because I thought that se doesn't have any preposition. I found out that se på has a separate meaning from se. From what I understand it means consider, therefore this sentence would be better translated I will neither consider it nor think about it.
Well in this case, 'se på' actually just mean 'to look at'. You can use 'se' without a proposition as well, as you say. For instance:
'at se fjernsyn/tv' = to watch tv
'kan du se solen?' = can you see the sun
'se på mig' = look at me
My first language is English and this sentence is fine as is in English, but one can also say I will neither see it nor think about it. That version conveys a stronger feeling about whatever you don't want to see or think about. Grammatically they are both correct.