By a quaternionic plane curve I mean the zero locus of a noncommutative polynomial in two variables, $x$ and $y$ say, over ${\Bbb H}$, Hamilton's quaternions. It is evidently well-known that, after identifying ${\Bbb H}$ with ${\Bbb R^4}$, one can capture with a single quaternionic polynomial any locus that occurs as the set of solutions of four real polynomials. (One needs a trick: if $Q=a+bi+cj+dk$, $(-Q+iQi+jQj+kQk)/2=a$; and similarly for $b$,$c$ and $d$).
Is there a criterion in terms of ${\Bbb H}<x,y>$ for two polynomials to cut out the same zero locus? Of course in principle one could unwind everything and then appeal to some appropriate realnullstellensatz, but I'm interested in a solution that is idiomatically quaternionic.
And a softer question. Capturing a real 4-fold in ${\Bbb R}^4$ as a specific quaternionic plane curve singles out a way to produce the 4-fold as the intersection of 4-hypersurfaces. That imposes a great deal of extra structure on the 4-fold. Indeed presumably too much, seeing as certain natural quaternionic manipulations (multiplication by non-zero constants on either side) change the intersecting polynomials but not the locus. What is the right level of extra structure in order to move to the quaternionic picture?
