In various texts and articles I've read which take aim to define properties in their most abstract and general terms, without reference to any particular sets or fields, the inner product is defined on a vector space with three conditions: left-linearity, positive definiteness, and conjugate symmetry.
When I in turn seek to define the property of conjugacy in the same abstract sense, these same sources tell me that it isn't a rigorously defined property in mathematics, and that it depends on a context and special applications, wherein conjugacy is defined. Often times in reference to inner products, conjugates are defined in the very familiar complex number space most of us would think of... the conjugate self-symmetric reals or the complex numbers are used in practice, but we need not define a conjugate in this typical way, nor are we required to define conjugates on real or complex numbers at all but on general elements of some other space. Im fine with this, I suppose.
But I come back to the definition of inner product, and am left wondering what kind of conjugate is being referred to. Does it matter? Do the authors of these articles have it in their head on some level they're talking about complex numbers? Maybe any definition of conjugacy can be used and it really doesnt matter. But why is the condition in the definition at all?
What is so special about an otherwise undefinable operation (i.e. conjugacy) that makes it a required condition for another definable operation (i.e. inner product)?