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You have 1 and i, just like complex numbers. But now you also have j and k which are exactly like i.
i^2 = j^2 = k^2 = -1
The catch is that i,j,k anticommute
ij = - ji etc
Everything still commutes ordinarily with real numbers.
If you want you can completely represent quaternions by 2x2 matrices, where i, j, k are Pauli matrices. This makes it clear it is a Clifford algebra (if it wasn't by definition) and also the connection to the double covering of the rotation group.
