Can you suggest PCB fabricators that have experience with complex multi-layer board designs?

Yes. If by complex multilayer you mean things like 8–30+ layers, HDI/microvias, controlled impedance, high-Tg or low-loss laminates, fine-pitch BGAs, or rigid-flex, there are several fabricators worth putting on your RFQ list.

Fabricators to consider

• Altus PCB — Cresskill, NJ. A regional option relatively close to Long Island, focused on higher-complexity PCB sourcing/fabrication and engineering support.
• Pad2Pad — Mahwah, NJ. Offers prototype-to-production fabrication and supports boards up to 32 layers, including rigid, flex, and rigid-flex.
• AdvancedPCB — Particularly relevant if you need HDI, RF/microwave, flex/rigid-flex, or a combination of fabrication and assembly. It describes capabilities ranging from standard through advanced PCB technologies.AdvancedPCB
• Altera Flex Circuits — Strong candidate for demanding rigid, HDI, flex, and rigid-flex work. Its published capability goes to 32-layer rigid boards, HDI/any-layer interconnect, and controlled impedance, with Class 3 capability.Alteraflex USA+1
• Calpak USA — Publishes capability for up to 30-layer boards, IPC Class 3, 2-mil traces, controlled impedance, RF materials, flex and rigid-flex.Calpak
• AtlasPCB — Focuses specifically on complex HDI, RF, multilayer and rigid-flex boards, with published capability up to 30 layers and stacked/staggered microvias.AtlasPCB
• SAYFU — Offers multilayer, HDI, flex and rigid-flex fabrication, with published capability up to 30 layers and explicit engineering-review criteria for difficult HDI stackups and microvia structures.SAYFU

If you're on Long Island

For local engineering/assembly support, there are also nearby companies such as Geometric Circuits Corporation in East Setauket and A A Technology Inc. in Ronkonkoma. These are more relevant if you want local PCB/PCBA engineering and manufacturing support rather than simply sending a complex bare-board design to an overseas fab.

For a genuinely difficult board, I'd compare vendors on stackup engineering, HDI experience, impedance control, via-in-pad/filled microvias, registration capability, material availability, IPC Class 3 qualification, and DFM support, rather than simply maximum layer count.

If you tell me your layer count, board size, trace/space, minimum via, material (FR-4/Rogers/etc.), impedance requirements, and approximate annual quantity, I can narrow this to 3–5 fabricators that specifically fit your design.