As in, personality? Physical dimensions? A character who can travel to different dimensions? Or a character who exists in multiple dimensions?

There's a few ways to view that question, so I'd suggest clarifying what you want.

PERSONALITY.

Well, obviously if you want them to be a multi-dimensional or a three dimensional character, it means making them more than a two-dimensional one.

For example, if a character's personality and motivation ranges from "Protect the princess" to "Be an arrogant jerk while doing so" and perhaps "Ask for a kiss from said princess every time you protect her" then you have the 80's DiC cartoon version of Legend of Zelda's Link. Or replace the kiss with "eat chilli dogs" and you have DiC's version of Sonic The Hedgehog's Sonic.

If we're going on cartoon characters, as they are the most 2d archtype of character in both how they are drawn and written, a good show which gives off how the characters develop is Gargoyles. In that show, there are beautiful subtleties, such as two character who admit to not being able to read being in the background of some scenes slowly wording out a book together, or one who gains a grudge against firearms going out of his way to break them anytime he sees them. One who has been betrayed so often that he gets a lone wolf streak about him. The impressionable one emulating cinema heroes that he sees etc etc.

The word develop is, by the way, the key to making a character 3d. If they are the same person from start to finish, they haven't been effected by the events of anything happening to them.

I would like to say this is not a rule, however. The Simpson family, particularly Lisa, do have character development, despite a general assumption that there is no common continuity to the show.

So, let's put this in terms of Sonic.

Let's say that Sonic starts as the hedgehog we know and love. First off, his biggest character quirk which a writer could play with and develop him on, would be his fear of water. Sonic X actually had a good episode which had him on a boat with Chris' family, trying to escape as he craves running and freedom, and fears water.

Now, that's fairly shallow character, but you can understand WHY he is attempting to do this. He loves to run, he hates water. It makes sense his character would treat a boat as torture.

That's a small example, but if you write a character, be sure to give them a personality. Not just "they are bright, except when it gets serious", but let the audience empathise with them. Whether it be building a backstory which shows why the character is the way they are, having a story which tries to tackle a flaw or expose a weakness to explore into their heart or just to come across something new and unexpected and then create a reaction which is more than the normal.

...

I dunno. It's hard to describe. I hope that helped.

I feel that multidimensionality is less to do with creating a character and more to do with writing a story. It's all down to their personality, how the interact with people, how they react to events and how they impact their lives. As Craig says, it's a difficult thing to quantify, but the 'easiest' way if there is such a thing is to have them develop over time.

If you want to create a character, I always find it's best to create the character how you want them, basically an outline of what traits you want them to have, and then decide work out their backstory; what did they used to be like? What events happened in the past to make them what they are now? How did they react to them at the time? Try not to think about fleshing them out on a canonical future to start with, as if you start with a character you like you will likely not be eager to change them for the sake of characterisation, whereas if you take that character and work backwards, at least then you now, when you put it in chronological order, you'll be working towards a state you can be proud of.

Evolution is the key