No, i, j, and k are not the same thing. They are the three distinct square roots of -1. You thus wind up with a space with one real axis and three orthogonal imaginary axes.

Why should -1 have three distinct imaginary roots? Well, why should it have one? Essentially, we just made up i, and we found out that the complex numbers had some really useful algebraic properties. The same is true of the quaternions.

But why not two imaginary roots, or four, or 17? Those turn out not to have nice algebraic properties. The only other thing out there are the octonions, with 7 imaginary roots.