Functions without complex roots, but with quaternion roots Many introductions to complex numbers begin with the question "What are the roots of $x^2 + 1 = 0$?" This function does not have real roots, but does have complex roots.
Are there functions which, in a similar vein, do not have complex roots but do have roots in the quaternions?

perhaps surprisingly, the answer here mostly depends on what you consider to be a function (and there are several choices that are more or less valid).

*

*a mapping of input points to output points. in this case the question isn't especially interesting because you can construct a function that has or doesn't have roots in any region.


*a continuous function defined using only complex coefficients. here there still are examples e.g. xy-yx=1


*analytic functions. this is a broad class of functions which includes all polynomials, logs, and a bunch of other stuff. I'm not sure the answer in this case.
