How can $i^2 = k^2 = j^2 = ijk = -1$ be true? I have just started to learn the basics of quaternions, but I immediately run into a wall.
Litteraly the first equation on Wikipedia is as follows
$i^2 = k^2 = j^2 = ijk = -1$
This implies
$i = \sqrt{-1}$
$j = \sqrt{-1}$
$k = \sqrt{-1}$
but now $ijk = -1$ also need to be true 
$\sqrt{-1} * \sqrt{-1} * \sqrt{-1} = -1$
$-1 * \sqrt{-1} = -1$
$\sqrt{-1} = 1$ 
This can not be true. What am I missing here?

The way to think about this is not to think of these as normal multiplication, but rather rotation. To rotate by i means to take the point at 1 and sort of move it 90 degrees up to i. Rotation by j and k is completely similar. All other numbers on the unit circles of i, j, and k for their respective multiplications follow the same 90-degree rotations. What happens to the other quaternions? Well, the unit circle of j and k, for the rotation of i, follows a similar motion, where j goes to k, and k goes to -j. And, for multiplications by other quaternions, you can probably guess what happens by now. Obviously, 180 degrees of rotation starting at 1 gives -1. Now we can tackle ijk. As I already said, ij gives k. This means that ijk = k^2. Obviously, this is -1. Problem solved!
