Why is quaternion algebra 4d and not 3d?

Why is quaternion algebra 4D and not 3D? Complex algebra is 2D and what is known as quaternion algebra jumps to 4D.

$i^2 = j^2 = k^2 = ijk = -1$

Using $1, i, j,$ and $k$ as the base (where complex uses $1$ and $i$ (or $j$ if you are an EE)) which results in a 4-axis space. Why is there no 3D algebra?

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possible duplicate of Is there a third dimension of numbers? –  Guess who it is. Oct 12 '11 at 17:10
This is related... –  Guess who it is. Oct 12 '11 at 17:11

@Henning It is not too difficult to show that a 3D-division algebra over the reals cannot exist. If $D$ were such a beast, and $x\in D, x\notin\mathbf{R}$, then consider the left regular representation of $D$ by 3x3 real matrices. The matrix $A$ representing multiplication by $x$ from the left cannot be scalar, because $x$ was not one. OTOH the eigenvalue polynomial of $A$ is cubic, and hence has a root $\lambda\in\mathbf{R}$. We have just shown that $x-\lambda$ is not invertible. –  Jyrki Lahtonen Oct 12 '11 at 15:46