  It is well known that Clifford (geometric) algebra offers a geometric
interpretation for square roots of -1 in the form of blades that square to
minus 1. This extends to a geometric interpretation of quaternions as the side
face bivectors of a unit cube. Systematic research has been done [32] on the
biquaternion roots of -1, abandoning the restriction to blades. Biquaternions
are isomorphic to the Clifford (geometric) algebra $Cl(3,0)$ of $\mathbb{R}^3$.
Further research on general algebras $Cl(p,q)$ has explicitly derived the
geometric roots of -1 for $p+q \leq 4$ [17]. The current research abandons this
dimension limit and uses the Clifford algebra to matrix algebra isomorphisms in
order to algebraically characterize the continuous manifolds of square roots of
-1 found in the different types of Clifford algebras, depending on the type of
associated ring ($\mathbb{R}$, $\mathbb{H}$, $\mathbb{R}^2$, $\mathbb{H}^2$, or
$\mathbb{C}$). At the end of the paper explicit computer generated tables of
representative square roots of -1 are given for all Clifford algebras with
$n=5,7$, and $s=3 \, (mod 4)$ with the associated ring $\mathbb{C}$. This
includes, e.g., $Cl(0,5)$ important in Clifford analysis, and $Cl(4,1)$ which
in applications is at the foundation of conformal geometric algebra. All these
roots of -1 are immediately useful in the construction of new types of
geometric Clifford Fourier transformations.

Rings and Algebras;Complex Variables