You could use httpPost to post your complex object to your delete method.
E.g.
[System.Web.Http.AcceptVerbs("Post")]
public HttpResponseMessage DeleteComplexObject(Models.ComplexObject deleteme)
{
this.ComplexObjectService.Delete(deleteme);
var response = Request.CreateResponse(HttpStatusCode.Accepted);
return response;
}
Here model binding is used to convert your json object into the ComplexObject automatically so you don't need to use [FromBody]