The problem with what you're trying to do is that you are taking a contradiction in stride and moving on as if nothing's wrong. This is not how the notion of numbers expands (from $\mathbb{N}$ to $\mathbb{Z}$ to $\mathbb{Q}$ to $\mathbb{R}$ to $\mathbb{C}$). We don't define "new" numbers that have, as you say, impossible properties, but numbers which extend the definitions of existing operations to domains where they were previously undefined.
For instance, working with natural numbers, the difference $a-b$ is undefined if $b\gt a$. Defining it for all values of $a$ and $b$ gives rise to the concept of integers. In a similar way defining $a \over b$ when $a$ is not a multiple of $b$ gives us rational numbers, and so on. This practice is called closure, and when we do it, we say the resulting set of numbers "closed" under the operation in question.
What you're doing is simply declaring that a pretend "number" behaves in a certain way in some circumstances and trying to then determine how that number would then behave in other circumstances. There is nothing wrong with that, per se, in fact people do it all the time, but they don't call the things they invent "numbers" because they aren't numbers. They don't have the properties numbers have or behave the way numbers behave. When you do this sort of thing, though, you need to make sure that the way you define your new entity is clear and consistent. You seem to have a problem there. For instance, since your made up entity $j$ interacts with numbers, you might want to think of it as an operator that works on numbers, letting $jk=k^2$ but how then do you define $j^n$? If you define it recursively as $j^{n+1}k=j(j^nk)$, you have a problem since $j^2k=j(jk)=j(k^2)=(k^2)^2=k^4\neq k^3 =j^2k$ by your original definition. Thus $j^n$ and $j^{n+1}$ as you define them are related in a way that is not really consistent with our usual understanding of exponentiation when it comes to operators. Thus you may want to use a different notation (say $j_n$) to designate this related family of operators.
Whatever you do, though, $j$ is not a number, so $jk$ is not really the product of $j$ and $k$, but the application of $j$ to $k$. it makes no sense to try to raise a number to the power of an operator. The result is undefined, and unlike the cases above, it's undefined for all numbers and all $j_n$, so you have no analogy to work with to "close" it. You could essentially define it any way you like, and it would be an entirely new operator. Thus, your question about $e^j$ is essentially meaningless.
