I admit that the term "complexification" is rather ill chosen and I'll gladly see it replaced by any other denomination. The term occured to me in the context where I had the impression that the quaternion algebra is a sort of "complexification" of the field of complex numbers. This is why I define the complexification of a field $F$ equipped with a conjugation, an automorphism $F \rightarrow F: z \mapsto \bar{z}$ of order $\leq 2$ (possible the identity), by defining a multiplication on $F \times F$ by the equation $$ a, b, c, d \in F \qquad (a,b)\cdot (c,d) = (ac-b\bar{d}, ad + b\bar{c})\quad (*) $$ Together with the pointwise addition this forms a ring structure on $F \times F$. This is a formal construction of the same less formal one of adjoining an element $\mathbf j$ to $F$ subject to the rules $\mathbf j^2 = -1$ and $\mathbf j a = \bar{a}\mathbf j$. For $F = \Bbb R$ with the trivial conjugation this gives the complex numbers and for $F = \Bbb C$ with the complex conjugation we get the quaternion algebra. Let $K$ bet the complexification of $F$ then the elements of $K$ can be written as $a+b\mathbf j$, and a conjugation of $K$ can be defined by $\tilde{z} = \widetilde{a+b\mathbf j} = \bar{a} - b\mathbf j$ with the property $z\tilde{z} = a\bar{a} + b\bar{b} \in F$. From this we see that if the conjugation is trivial then the ring $K$ is commutative. The first question that arises is the existence of zero divisors. We can write $(*)$ in matrix form $\left( \begin{smallmatrix} a & b\end{smallmatrix} \right)\left( \begin{smallmatrix} c & d\\ -\bar{d}&\bar{c}\end{smallmatrix} \right)$ so that the (right) zero divisors $z = c + d\mathbf j$ are characterized by the propery $z\tilde{z} = 0$.
Some examples: Apart from the trivial examples $\Bbb C$ and $\Bbb H$ we can construct examples where the field $F$ is taken with the trivial conjugation. The condition for zero divisors then reduces to $\exists a,b \in F: a^2+b^2 = 0$ This results in commutative rings $K$ which are either fields or direct sums of fields. Another series of examples can be obtained for fields whose automorphism group contains an element of order $2$, this includes the cyclotomic fields, and all finite fields of order $p^{2k}$. I would like to know if it is worth while to study those examples or if all is already known about those constructions. It has to be noted that this construction is not the same as the the quaternion algebra over a field $F$. For example if $F = \Bbb Q(\zeta_3)$ then $K$ has dimension $4$ while the quaternion algebra associated with $F$ has dimension $8$ over $\Bbb Q$. For finite fields of order $p^{2k}$ the Galois group has order $2k$ so contains an automorphism of order $2$, but the examples I tried all have zero divisors and I have no idea about their structure (in the light of the Wedderburn theorem).