What are the most exciting mathematical structures that were recently discovered? I hope this question doesn't come as off-topic, but I'm a programmer and I've been deeply bored with how mundane are the structures I interact with daily. Arrays, lists, real numbers... the most interesting things are perhaps complex numbers, quaternions, some graphs and cellular automata. All those felt mind-blowing when I first found about them, but now, they're just normal.
I wonder how many "quaternions" I'm missing for not being a mathematician. What are some amazing mathematical structures that were recently discovered?
 A: Disclaimer: technically I'm not answering your question, because I have used the word "structure" in a more general way than I believe you intended. Also I feel like talking about some maths I love.
I can't say much about "recently discovered", but there are two main approaches to adding "structure" to sets: an algebraic approach and an analytic approach. The former is based on defining binary operations - e.g. groups, closed under addition with a few extra properties, and fields such as $\mathbb{R}$ which have addition and multiplication as defining operations. This is what is commonly referred to as a "structure".
However, I'm more of an analysis person so I prefer looking at "spaces" (which is generally the term used for the analytic approach to adding structure to sets). My research is essentially studying certain subsets of the hilbert cube. This is basically an infinite dimensional cube. Recently however, I've just been focussing on subsets of the plane.
I could never have imagined these to be so complicated. A "continuum" is defined to be a compact, connected metric space. If you don't know what these terms mean - a "metric space" is anything that takes up space - there's a notion of distance between distinct points in the set. "Connected" is what you think it means. A circle in the plane is connected, but two distinct parallel lines are not. Finally "compactness" is difficult, but it's like the "continuous equivalent" of finiteness. There's a brilliant description here.
Anyway, "continua" are these single objects which are nicely self enclosed. Hopefully I'm making sense.
An obvious question is: Is it possible to classify the continua that are subsets of the plane? (aka plane continua.) A huge number of people have studied plane continua and yet we're no where near any sort of classification. One class of continua that have been studied extensively are "indecomposable continua".
Try and visualise a continuum. This is anything like a blob, or an elastic band, or any singular object that contains its boundary. Clearly (whatever it is you've imagined) it's possible to split it up into smaller continua, right? You can find two subcontinua of your original continuum so that the union of them is equal to your original continuum.
Well, it turns out that this isn't necessarily the case. An indecomposable continuum cannot be realised as the union of any two of its proper subcontinua! The existence of such continua is already counterintuitive, but it was even shown that "most" plane continua are in fact indecomposable.
I've talked enough now. All of this stuff is a subfield of "topology", the field of maths where mugs are no different to doughnuts. A "topological structure" basically has the least amount of structure possible while still being interesting. (I guess it's the opposite of what you asked then!)
A: Try studying infinite numbers.  They will be quite different from anything in software.  (Though they aren't particularly new.)
You could look at octonians.
A: With respect to quaternions you mention, there were some exciting new numbers, called periods discovered just a few years ago.

The definition of periods below is from the fascinating introductory survey paper about periods by M. Kontsevich and D. Zagier.
Periods are defined as complex numbers whose real and imaginary parts are values of absolutely convergent integrals of rational functions with rational coefficient over domains in $\mathbb{R}^n$ given by polynomial inequalities with rational coefficients.

The set of periods is therefore a countable subset of the complex numbers. It contains the algebraic numbers, but also many of famous transcendental constants.
Note: One nice aspect is that the famous equality $$\zeta(2)=\frac{\pi^2}{6}$$ can be shown based upon the fact that $\zeta(2)$ and $\frac{\pi^2}{6}$ are periods which form a so-called accessible identity. This is shown in this answer which also is based upon the cited paper.
