I prefer 'flatter than rounder' necks, and a flatter radius than a shorter. I like basses in colors, and they got to look right: I can't be playin' no ugly axe.
Reasonable weight, and more compact if possible: I loved the Alembics, but in long scale, they were l-o-n-g instruments. I once walked in a store with one in its case and the guy asked me (before I opened it) when I started playing keyboards . . . . .
I'm one of the relatively few that really prefers true active pickups and controls (NOT hi-z pickups ganged to active EQ), I prefer the quiet and generally baked in great tone. I'll admit I've been spoiled rotten at this by real EMG setups, much less the Alembics.
But past that, the famous Three Question Test ya got to pass to come home with old JW:
1) It needs to have a reasonable feeling neck and fingerboard that will accept a setup and keep it for a reasonable period of time. DON'T want to be cranking on it every 2 weeks. I generally prefer necks with a rosewood, pau ferro, or ebony fingerboards for the generally added stiffness an extra piece of wood added to the neck can supply, especially ebony.
2) It's got to get in tune and stay in tune for a reasonable amount of time (like I tuned it at 9 and it BETTER still be in tune at last call, which mine usually are). I'm NOT gonna stand there and be one of those dorks that's ALWAYS TUNING A BASS FOR FIVE HOURS. NOT gonna HAPPEN ! So I'm looking for good keys, a proper nut, an accommodating neck, and a good bridge.
3) It's got to be quiet electronically. I don't care if it's single coil or anything else, the only noise I want coming from it . . . . is me. Not WSM and the Opry. No crackling pots, no hum when I take my hands off it, nothing. I will allow for the occasional 'I'm next to the dimmer pack and there's an arc welding shop next door' buzz that nothing could get around, but otherwise, Hey! I want it quiet in here !
Because if I've got to fight bad action, I'm chasing keeping it in tune more than I should, and it's buzzing, it's useless for me, regardless of the price point. Firewood. Door stop. Hood prop.
And ultimately, these things rearing their head on stage or recording is just completely unprofessional, and real pros won't keep you around long if this is ever more than a once in a VERY great while occurrence.