String theory assumes that strings are complex geometric objects

A string is a 1 (real) dimensional object - it is simply an extension of a point into one higher dimension.

A string theory describes how a string behaves in an ambient spacetime of some higher dimension through the embedding of the string into this ambient spacetime. This embedding results in a two real dimensional surface (called the worldsheet) with complex structure which is roughly thought of as the string+its history (it is exactly analogous to a worldline). Because of the complex structure it makes sense to say the string worldsheet is 1 complex dimensional.