I'm trying to create a Mandelbulb in Blender 3.3.1 using the scripter. Zero errors but nothing happens when I run the code

There are two little mistakes in your script as Gorgious wrote in the comments:

1. the operator properties "size" and "iterations" must be defined by ":" not by a "="
2. bpy.props.IntProperty has no precision parameter.

import bpy
import numpy as np
import math

def mandelbulb(x, y, z, iterations):
    c = x + y*1j + z*1j
    z = c
    r = 0
    for i in range(iterations):
        r2 = x*x + y*y + z*z
        if r2 > 2:
            return i/iterations
        theta = math.atan2(math.sqrt(x*x + y*y), z)
        phi = math.atan2(y, x)
        r = r2**(1/8)
        x = r * math.sin(theta/8) * math.cos(phi/8) + c.real
        y = r * math.sin(theta/8) * math.sin(phi/8) + c.imag
        z = r * math.cos(theta/8)
    return 0

def generate_mesh(size, iterations):
    vertices = []
    faces = []
    for x in np.linspace(-2, 2, size):
        for y in np.linspace(-2, 2, size):
            for z in np.linspace(-2, 2, size):
                value = mandelbulb(x, y, z, iterations)
                if value > 0:
                    vertices.append((x, y, z))
    return vertices, faces

def create_mesh_object(vertices, faces, name):
    mesh = bpy.data.meshes.new(name)
    mesh.from_pydata(vertices, [], faces)
    mesh.update()
    object = bpy.data.objects.new(name, mesh)
    bpy.context.collection.objects.link(object)

def execute(size, iterations):
    vertices, faces = generate_mesh(size, iterations)
    create_mesh_object(vertices, faces, "Mandelbulb")

class MandelbulbOperator(bpy.types.Operator):
    bl_idname = "object.mandelbulb_operator"
    bl_label = "Mandelbulb Operator"
    bl_options = {'REGISTER', 'UNDO'}

    size: bpy.props.IntProperty(
        name="Size",
        default=32,
        min=1,
        max=256,
        step=1
    )
    iterations: bpy.props.IntProperty(
        name="Iterations",
        default=64,
        min=1,
        max=512,
        step=1
    )

    def execute(self, context):
        execute(self.size, self.iterations)
        return {'FINISHED'}

    def draw(self, context):
        layout = self.layout
        layout.label(text="Create a 3D Mandelbulb")
        layout.prop(self, "size")
        layout.prop(self, "iterations")
        
def register():
    bpy.utils.register_class(MandelbulbOperator)

def unregister():
    bpy.utils.unregister_class(MandelbulbOperator)

if __name__ == "__main__":
    register()

After you run the script, no error should be displayed in the System Console window and nothing will happen because you only have registered the operator Mandelbulb Operator.

Use the main menu Edit > Menu Search... F3 or Edit > Operator Search... to find your operator. Both should work.

Menu search:

Operator search:

Select it and hit Enter to execute it.

With the Adjust Last Operator popup at the bottom left of the screen you can change the settings Size and Iterations of your Mandelbulb Operator.

If for some reason you cannot see the popup make sure the option View > [X] Adjust Last Operation is ticked in the viewport's menu, or press F9 to bring it back.