"I want to have my own apartment."
Translation:Chcę mieć własne mieszkanie.
So that means even though "chcieć" takes the Genitive case, "własne komputer" here is actually an object of "mieć" and hence it's in the Accusative.
Does that imply the sentence "I want to look for an apartment" would translate to "Chcę szukać mieszkania", since "szukać" takes the Genitive?
I think that elsewhere, it was stated that it sounded rather impolite to omit the full verb succeeding the auxiliary verb. Otherwise, I for myself would have argued that omitting the full verb after the auxiliary verb sounded simply incomplete, as if you were still a learner of the language, deriving from a completely different language stem, thus unfamiliar with the details of the language in which you just spoke. In Germany, it's a feature spot frequently with migrants from Turkey or the Middle East, when certain parts of a sentence are involuntarily omitted because they yet not spoke German fluently. They would omit prepositions as well as verbs when accompanied by auxiliary verb. Thus, I would argue that you should still add it to your sentence, although you may by now have answered this question for yourself.