Do complex numbers behave like ordinary numbers?
Yes and no.
Complex numbers form a field - that is a set equiped with addition and multiplication that behave in the ways with which you are familiar. In technical language, a field is an abelian group under addition, its non-zero elements form an abelian group under multiplication and multiplication is distributive over addition.
Like the rational numbers, which it contains, it is a field of characteristic
This leads to the Complex numbers behaving much like the rationals or Reals when simply treated as numbers.
For example, you can use the quadratic formula directly to provide the roots of
There are some interesting things that happen with radicals and Complex numbers.
For example, we are used to thinking that Complex numbers are naturally expressible in the form
Note however, that if you are happy to just think of Complex numbers as numbers, leaving expressions like