Numbers of the form $\{z= x+ i\,y:\;x,\, y\in\mathbb{R}\}$ where $i^2 = -1$. Useful especially as quantum mechanics, where system states take complex vector values.
Complex numbers - together with multiplication and addition - are a field of numbers of the form $\{z= x+ i\,y:\;x,\, y\in\mathbb{R}\}$ where $i^2 = -1$. In physics they are a useful representation of quantities that have magnitude and phase such as quantities that vary sinuosoidally with time. System states in quantum mechanics live in a complex Hilbert space usually with a countably infinite basis but sometimes of finite dimension.
The complex numbers with addition and multiplication are the smallest algebraically closed field containing the ring of integers with addition and multiplication, i.e. they are the smallest field needed to solve any polynomial equation $p(x)=0$. They are also the largest connected, locally compact, topological field: intuitively - the biggest field with "everyday" continuous arithmetical operations.