Maybe my question is rather weird, but actually it turned out it is also a question of my colleagues. Usually when you register for an academic conference, there's an option that you can register a companion. I'm wondering if it is appropriate to register my girlfriend as a companion? Does it sound so bad?
In the conferences I am aware of, companions are people not attending the conference, but coming along with some of the participants for tourism or company. Depending on the venue, they may or may not offer some alternative activities for them (anything from visits to the city to a heap of maps and sort yourself out).
If this is the case, your girlfriend would fit right in, as long as she is not interested in the conference itself.
Davidmh's answer makes sense for conferences where registration is a "package deal" including lodging, meals, maybe travel, etc. Here is another possibility.
At the annual Joint Mathematics Meetings, participants have the option to register a "nonmathematician guest" for a small additional fee. The conference describes this as:
Any family member or friend, who is not a mathematician, and who is accompanied by a participant in the meetings is eligible for this category. Guests will receive a badge and may accompany a mathematician to a session or talk and may also enter the exhibit area.
(The "exhibit area" in this case has booths operated by publishers, software vendors, etc, often selling their products at a discount.)
For this conference, registration doesn't include any travel, lodging or meals (participants are supposed to arrange those separately), so the only benefit you get from registering a guest is that they get a badge to attend the conference. If you are traveling with a companion but they don't want to get in to the conference, there would be no reason to register them at all.
Some possible reasons to do this:
You need someone's assistance to get around inside the conference (pushing a wheelchair, carrying books, etc)
Your guest is curious what a conference is like
Your guest wants to be able to meet you inside the conference venue during breaks, meals, etc.
You just want to be with your guest during talks, etc.
The "nonmathematician" requirement is to point out that this is not supposed to be used as a way for you to get a colleague into the conference for cheap (at this conference, registering a guest is $16 while regular registration was $250-$400). Of course the only enforcement would be the honor system.
So a girlfriend would be just the sort of person for whom this category is intended, provided that she is not also a colleague.
I once made use of this option when the conference was held in the city where my parents live. My father is not a mathematician but is interested in math, so I registered him as my guest. He enjoyed attending some of the less technical talks, browsing the books in the exhibit hall, and generally just experiencing the atmosphere of a mathematics conference.
The companion option provides a convenient package allowing someone to participate in the nonscientific aspects of the conference. For example, I've been to some remote conferences where if your guest wasn't a conference attendee, meals had to be eaten at the very expensive facility restaurant, instead of at the conference buffets.
These kinds of things can be very convenient. I've seen people who thought they'd be able to avoid the cost of companion registration, only to knuckle down and buy it at the "gate price" once they're on the ground at the conference, and this can be more expensive.
Sometimes, companion registration may be an option because it is anticipated that non-conference attendees may have a rough time without it.