あるいは details
Examples
or; alternatively — two candidates are set side by side and either may hold
ご質問は電話、あるいはメールでお寄せください。
go shitsumon wa denwa, aruiwa meeru de o yose kudasai.
Please send your questions by telephone or by e-mail. formal
この問題は法律の面から、あるいは経済の面から論じることができる。
kono mondai wa houritsu no men kara, aruiwa keizai no men kara ronjiru koto ga dekiru.
This problem can be discussed from a legal angle, or alternatively from an economic one. neutral
possibly; it may even be that — heading a clause the speaker puts forward as one live possibility
あの様子では、あるいは彼はもう知っているのかもしれない。
ano yousu de wa, aruiwa kare wa mou shitte iru no kamo shirenai.
Judging by how he looked, it may well be that he already knows. neutral
交渉は難しいが、あるいは来月中にまとまるだろう。
koushou wa muzukashii ga, aruiwa raigetsu chuu ni matomaru darou.
The negotiations are difficult, but they may possibly come together within next month. neutral
Formation
AあるいはB（電話あるいはメール・辞書あるいはネットで調べる）attaches to noun phrase (schematic)あるいは、clauseattaches to plain clause
Notes
sets up the guess and something at the end of the clause has to complete it.
Related patterns
Similar
それとも needs questions on both sides and asks the listener to pick; あるいは states the alternatives itself and is at home in a written argument, where handing the reader a choice would be out of place.
The same "either will do", one register apart — または is the neutral written default and the one a form or a notice uses, while あるいは carries an essayistic tone and can additionally soften into "possibly" (あるいは間に合うかもしれない), a reading または never has.
Lookalikes
Both head a clause that ends in かもしれない, and they part on register and strength — もしかしたら is the spoken "maybe, just maybe" and takes the speaker by surprise, while あるいは weighs a possibility the writer has already been considering.
Combines with
In its "possibly" reading あるいは wants the sentence to end in かもしれない or だろう — it colours the whole clause as a guess and cannot carry that guess alone (?あるいは中止になる).