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DEEP & SURFACE LEVEL ATTRACTION
On the surface level, I feel like Edward would be very attracted to someone he perceives as a riddle or mystery to be solved (like with Jessica in Joker’s Asylum) or someone he idolises (like Bruce / Batman in Batman: Forever and Batman 2022). Of course, attraction based on just either of these factors can easily fade once Edward believes he ‘has’ you or finds out you’re not the amazing person he thought you were. At which point, the relationship is likely to be violently terminated, meaning he either tries to hurt you (Batman: Forever and 2022 once again) or one of your loved ones (he shoots Jessica’s father dead in Joker’s Asylum).
Furthermore, he’d be easily driven to pursue someone who ‘belonged’ to Bruce / Batman as part of his efforts to show him up (as he does in Batman: Forever). However, none of this constitutes as actual love.
He’s keenly aware of when a person (a woman specifically) is trying to use him, as he states in Batman: Prelude to the Wedding: “I have money… I have power… Because of this, some women found me attractive. A certain kind of woman anyway.”
Long story short, shallow connections can only result in sexual relations “simple animal urges. Raw physicality” (as the Edward above puts it) and violence.
A deeper connection can only be forged with Edward if he believes you to be his equal, or at least something close to it. I think it’d definitely help to start off platonically, so he doesn’t become obsessed and eventually tire of you (see above).
His relationship with the Penguin in the Gotham TV show is probably the best example of a genuine relationship (albeit, I haven’t seen the show in full, so my analysis may be limited) one could have with him. He’s extremely devoted to Oswald after they fall in love, not to the violent extent the other is, but still.
EDWARD & A MALE PARTNER
Gotham is also the perfect example of what Edward would be like in a relationship with a man. In terms of sexuality, I’d likely pin him as bisexual with a preference for men, though he’s very likely repressed or ashamed of it in many versions due to the time period (even if the homoerotic tension he has with Bruce is off the charts in 1995’s Batman: Forever).
And before you argue with me that Edward is too smart to suffer from any kind of internalised homophobia (or misogyny which I discuss later), bear in mind that intelligent people can be just as bigoted as idiots sometimes, especially when said bigotry is acceptable in their time period. Like, Edward is by no means a manly man, I’m sure he’s been accused of being gay at least once in his life (talking generally here) due to his effeminacy or intelligence or something like that. Even if he’d know that being gay isn’t actually bad, being accused of it would at the very least cause him to resent his attraction to men, even just the littlest bit.
More about Edward with a male partner, I think he’d be more likely to challenge him either via riddles (as he does with Batman in literally everything ever) or just generally challenge his authority. At first, Edward is intimidated by his feelings for you, so he attempts to hide his affections behind ‘proving his superiority’ or something alike. Maybe it's him yearning for the healthy male role model he never had, either due to his father’s shittiness or general absence.
EDWARD & A FEMALE PARTNER
With a woman, however, I think Edward would be less about challenging you (though he may challenge those he perceives as romantic rivals) and more about courting you. Though it doesn’t end well, his obsessive courtship of Jessica in Joker’s Asylum is a prime example of this (also Kristen Kringle in Gotham). He will write you weird, romantic riddles. He will send you gifts. He will learn everything about you. Once again, this sounds bad, but Edward would likely view you as someone to be won rather than challenged. Though, if this is a result of an underlying misogyny problem or if this is some weird attempt at winning the affections of the mother he never had depends on the version. It could very likely be both.
However, he isn’t above challenging his female lover if he believes her up to it, as seen in Batman: Prelude to the Wedding. However, this is generally a rarer case.
I also feel like he’d be pervier with a female partner (for the same reasons depending on the version, I might get more into this if I do kink headcanons for Eddie…). We see this in Catwoman: When in Rome when he’s always trying to touch Selina in some way or catch her when she’s in the buff. For most versions, though, I don’t see this as being overwhelmingly obvious (yes, even Zero Year Eddie).
GENERAL RELATIONSHIP HEADCANONS
I do think with any partner, though, he’d be condescending as hell. Even if he views you as his equal, no one is as smart as THE Riddler. He won’t be mean about it unless he’s in a bad mood, but sometimes it just comes out (I see this with Arkham him especially).
He’s bound to send you riddles in one way or another. BTAS Edward sends you them via hacking devices he knows you frequent, maybe even making computer simulations of your future life together (as seen in ‘What is Reality?’). Batman: Forever Edward sends you physical riddles, using pictures of your face as he does with Bruce in the movie. It never occurs to him how creepy this is. So on and so forth.
“A function of the brain confused with the heart, a reaction so strong it sets species apart. What am I?”
“Love.”
His crimes may even end up leading back to you, as they do his brief fancy with Batgirl in DC Superhero Girls (episode title is ‘Riddle of the Heart’ if you’re wondering). Albeit, he wouldn’t directly link you to them if you were a civilian, not wanting the Bat to some between you. He’d certainly use crime to get your attention if you were a vigilante, however.
How obsessive Edward is depends massively on the version, how much you reject him and how kind you are to him. All versions soak up your kindness and affection like a sponge, however, and typically ignore your rejections.
All versions probably stalk you, either physically or technologically or both. If or if not he kills people for you depends on how violent the version is overall (2022 would, BTAS probably wouldn’t unless he felt you were in danger), the same can be said about his willingness to kidnap you. A lot of it comes from his need for attention, he has a tendency to get jealous.
He’d be quite possessive, too. To quote him in Detective Comics Annual #8, “My story. My story. My story. MINE.”, but I don’t think his possessiveness over a partner would be extremely obvious unless he was angry or had genuine reason to worry about them. With Edward, you’re mostly free to do what you like, so long as it doesn’t involve any potential rivals.
If things are healthier between the two of you, he’s more likely to be protective anyways. He likes it, it makes him feel big and manly. Chivalrous. We can see this in Batman Confidential #27 in his protection of Carson from King Tut.
He’s somewhat affectionate, though some of it depends on the version and his mood. For the most part, he likes to keep an arm around you, especially when you’re out and about together. Having someone so gorgeous on his arm does both wonders for his ego and his sense of relationship security. He adores it when you return his affections, however; you make him melt when you kiss him or tell him how clever he is (BTAS ‘The Riddler’s Reform’).
Quite domestic, too. He loves it when you take care of him (more detail on this later). This is referenced in Batman: Forever when he says “I’m home, my sweet!” to Dr. Chase. Having a house-spouse (lair-spouse?) is right up his alley.
He can be romantic if he tries besides using his riddles (once again, DC Superhero Girls). If he really loves you, he’ll get you flowers, marry you, even.
In terms of pet names, “my dear” is always used. Though, you may expect to hear “my sweet” (Batman: Forever). I don’t think he’d be too crazy with the pet names unless he’s feeling particularly theatrical. Then, you can expect “my pretty”, “pretty lady”, “my cherub” and “my little (your name)” as he uses in the Batman ‘66 show.
EDWARD’S TYPE
Physically speaking, I don’t think he’d really have a type besides someone vaguely attractive. It’s really the personality that gets him. I think he’d really go for someone he believed was intelligent (there are too many examples for this) yet kind and understanding to him. You can see this in how he yearns for Batman’s approval in the 2022 movie, or how in Batman: Forever how he takes Sugar (Two-Face’s nice, normalish girlfriend) to the gala as opposed to the feisty, wilder Spice. He even remarks that Echo and Query are too wild for his liking in Detective Comics Annual #8.
He’d certainly appreciate someone who challenges him, though, just not so much that he gets too used to losing. Give him a challenge, but in the end, let him win.
His partner doesn’t have to be as smart as him, but I doubt he’d ever be attracted to someone he deemed ‘stupid’. Even if you struggle to answer his riddles, your enthusiasm in trying makes his heart sing. He just loves to see your mind work, the furrow of your brow… We kind of see this with Jasmin in Strange Love Adventures.
It would, however, massively give him the ick if his potential partner spoke ill of people he believes to be intelligent or his work. He has no patience for idiots, even nice ones, I’m afraid. Edward yearns for intellectual understanding almost above all else.
Because of his crazy daddy/mommy issues, I think any version would adore a partner who dotes on him and frequently fluffs his ego. He’s so used to feeling alone and unwanted, having someone who looks after him and shows they genuinely care would put him on cloud nine.
He’d really click with someone who knows what it feels like to be misunderstood or lonely or unwanted or whatever, especially if they had conditions that overlap with his own (see below).
MENTAL ILLNESS & DISORDERS
(Just wanna preface this bit by saying I don’t suffer from any of the conditions Edward canonically has - albeit there is some overlap between his OCD and my autism - and all of the research I’ve done has been online. I’m not an expert on this stuff, so I may be wrong about certain things).
Sometimes, I see people say that Edward would really struggle to love someone genuinely due to his canonical OCD, NPD and insanity (in some versions). And while I do think this is kind of true, I don’t think these factors would cause him to treat a partner of whom he had genuine feelings for badly.
Like, he’s (talking generally) very particular about his work and doesn’t want you touching or even interacting with any of it. He’ll get mad if you do so, but he won’t ever put his hands on you. Part of it comes from his desire to keep you safe, too; the most you can ever hope to involve you in his crimes is running his riddles by you to gauge how difficult they are.
Allegedly, when a person with NPD really likes someone they tend to put them on a pedestal, idealising them and project ‘infantile desires’ (unconditional love, admiration, comfort, etc) onto them. While this may not be very healthy, it’s one of the better outcomes you can hope for in a relationship with him (kind of seen in Strange Love Adventures).
It’s different to having him view you as a ‘riddle to solve’, too (see beginning of headcanons). If Edward projects these desires onto you, he believes you to be a sparkling light of intelligence in this dull world - an angel, the very embodiment of love and comfort, all his.