I am looking for something more desirable than partner or amore in Italian. Of course amore is nice but it translates to love, but to say more indirect, is there anything else?
Maybe an expressions closer to "significant other" might be "la mia metà" (literally "my [other] half") or the variations "la mia metà migliore" ("my better half") and "la mia dolce metà" ("my sweet/lovely [other] half"). Those expressions are most commonly used to refer to the husband/wife but they are used also by unmarried couples, sometimes (but not necessarily) with a slightly humorous or ironic intent.
What one considers to be “nice” or “desirable” is of course highly subjective, but I find compagna/o the most usual and neutral-sounding word for a “significant other” (with whom one is not married). Other options include ragazza/o (more like “girlfriend/boyfriend”), which is most suitable for younger people, and fidanzata/o, literally meaning “betrothed”, but since few people get actually engaged, it's sometimes used for one's partner (and occasionally as an euphemism for a lover or some not-too-significant other).