That's not even close to all there is to it. Complexity analysis is a relatively young field with lots of deceptively simple unsolved problems.
For a simple example of how it's a lot more than that, follow Tarjan's proof of the disjoint set's amortized time complexity[0]. It's not at all obvious, even though the disjoint set is a very simple and practical data structure.
We’re saying the time, space, or some other metric of the solution scales with the size of the input number in a way that is not constant. The number could be any input (arbitrary).
That covers most of the algorithms you'll see in a coding interview test, but there's a ton more complexities outside of search spaces, sorts and nested O(n) loops.
Even some graph operations will leave familiar territory.
A matrix can always be split into groups of sub-matrix, and the groups of sub-matrix is itself a matrix. Applying Strassen algorithm recursively is therefore O(n^log2(7)) == O(n^2.8ish).
Thank you, but for the link provided it only says that these are the Fibonacci numbers and nothing more. Yes, you can click on this F(n) (how can you even know this is a link), but anyways.
That’s literally all there is to it.