Quaternions are the next step up in complex numbers. Complex numbers are real numbers with an 'imaginary' part. 'imaginary' is in quotes as that's what we call it, but it's not fiction, it's the square root of -1, often represented by i (or j in electrical engineering). Written differently i2=-1. In early days, people considered this strange and useless so imaginary is fitting. By the way, if you go back far enough people didn't see any point in 0 either.
A real number is represented by a - any real number like 2, 1.5781739, 10000, etc. A complex number is represented by a + bi where a and b are real numbers and i2=-1. Complex numbers are very good at represented rotations (as in a circle around 0) which is why they're important in many scientific and engineering activities.
Quaternions are the next step up: a + bi + cj + dk or a0 + ia1 + ja2 + ka3. Here i2 = j2 = k2 = -1. All good so far, but it gets more complex in that ij = k = -ji, jk = i = -kj, and ki = j = -ik. That leads to ijk = -1 which can be seen from ij=k and k*k = -1. The fact that ij = k and not -1 is a little confusing, but these are more than just the square root of -1, but directions in 'quaternion space'. What these look like is the outcome of a standard vector cross product of i x j = k and j x i = -k.
What's interesting about quaternions is that they have almost all the properties of real and complex numbers in terms of addition, subtraction, multiplication, division, and inverses, except that multiplication is not commutattive. In other words, for normal numbers a*b = b*a, but as we saw for quaternions ij = -ji.
In terms of their use, quaternions are useful for rotations in higher dimensions that planes where complex numbers are good. In many ways the 4 dimensional approach resembles relativity theory where time plus the three spatial coordinates are linked or common differential equations where time and space are linked - especially in Schrodinger's equation where time and space are linked via a multiplier of i. For fun purposes, quaternions can replace complex numbers to create higher dimensional Mandelbrot sets. Don't stop there - there are also Tessarines another 4-dimensional complex number were j2 = 1 and Octonions - an 8th dimensional complex number.
The story is that Hamilton was trying to solve a difficult problem and came up with the idea of triplets of complex numbers while out on a walk with his wife by the Royal Canal in Dublin. Supposedly, he was so struck by the idea that he carved i2 = j2 = k2 = -1 and ijk = -1 into the stone of a bridge.