I am teaching Differential Equations this semester. The pre-req is Calculus II, not even Multivariable Calculus. I made my peace with that, can't fight with the big-shots of the department. So, I have been teaching stuff from Multivariable calculus on a need basis (partial derivatives, chain rule, etc.) I also spent one lecture to teach basic matrix algebra, determinants. Then, evidently, I need a little bit of complex algebra as well, so I did that too. Now, I am teaching higher order differential equations, so kids need a bit further explanations for finding complex roots. Well, these kids will eventually take linear algebra, and multivariable. And it is the department's fault not to require these as pre-req. But what about complex algebra/variables? There is no calculus course, single or multivariable, where we introduce complex numbers and complex algebra. Why? What is the fix for this? I mean I am checking calculus books, syllabus from different universities and their calculus courses, no complex algebra whatsoever. How can this be?
Let me give you a slightly different perspective. I also come from a European background and have had the chance to TA and teach undergrad courses in the US. As a caveat I have not taught DE, but have taught many calc classes including multivariate and have graded for some DE.
First I understand your utter frustration if you come from Europe and are used to the European approach to teaching and the European undergrad students the US is a very very rude awakening. I remember being at one of my first calc I lectures and seeing the professor add infinite sums by terms without as much as a nod to convergence and present it as an ok thing to do and it was actually physically painful as I was clenching my fists in frustration that this is taught to someone. So believe me I understand that you want to teach the right stuff in the right way.
Second Xander is right. Noone is asking you to teach what you intend/try/are teaching. The expectation of the department and of the students and maybe most importantly of the follow up courses is that you will teach the students bags of tricks. The courses in the US do not generally correspond to similarly named courses in Europe.
Everything in the calculus sequence including DE is mostly about tricks and learning to handle the very basic formalism. Calculus students may be taught epsilon-delta proofs but 90% of them do not understand them and they are not really tested on whether they do. In the same way they are not taught real derivatives, no one cares about boundary conditions and pathologies and they are essentially never asked to think for themselves in the problems. Not really think anyhow I'm convinced that all the finals I've seen could be solved by a an easy program with minimal notational standardization.
There are reasons for this though (not necessarily good ones mind you). The students are generally not math majors and aren't going to continue on in math. Many of them have a fairly shaky background in math alltogether, I remember in calcII classes still consistently having students make elementary level errors such as sum of squares is square of sum and even splitting fractions in wrong ways.
All together if you try to teach complex algebra, (and other stuff you mentioned) along with DE you will be doing a disservice to your students. They will be expected to know the things that the department actually wants you to teach them and due to the extra stuff they will no have properly learned the bag of tricks that teachers further along will expect of them.
Complex algebra will be taught later on to those that continue down to math majors or physics majors (and possibly not even those. In some colleges/Universities many math majors require no advanced complex algebra course at all).
In the United States, high school students typically encounter complex arithmetic and algebra for the first time in the context of an Algebra 2 course, and then revisit the topics the following year in Precalc. (For most students this would be either 10th and 11th grade, or 11th and 12th grade.) Complex numbers also appear on the SAT and ACT.
The problem you are having is not that students haven't seen complex numbers before -- it's that they haven't seen them or used them recently. If you are teaching Diff Eq, most of your students are in their second year of college (if not later), which means that it has been roughly two years since they've had to work with or use complex numbers. As you point out, they do not play a role in single-variable Calculus -- but why should they? Calc 1 and Calc 2 are the study of functions of a single real variable; it's hard for me to even think of places where I could make use of complex numbers in those courses, even if I wanted to.
Complex roots of the characteristic equation are no big deal. Kids have seen that plenty with the quadratic formula in high school algebra.
I worry from your question that you are dragging the course down too much by feeling the need for prereqs and then diverting to quasi teaching them (matrices, partial derivatives, complex algebra, etc.) The vast bulk of ODEs can be taught without getting into that stuff. And the course is so full that you really have a hard time covering everything in a semester anyhow. Even when some tiny aspect of one of these things, say a determinant, is used, and it is not necessary to even cover that in a 15 week ODE course, you should just do the minimum, rather than seeing a need/opportunity to divert the class.