Buying a seat for the little one will give you the most comfort if you can afford it.
Having said that, I don't think its necessary when they're very small and not moving around much. 4 months is still young enough to use a travel bassinet for long-haul flights, but every airline has different policies on these. Some you may have to book (and pay for) in advance, others you may get free priority. Or turn up and keep your fingers crossed for a sympathetic move to a row with free seats. Check with the airline website.
We've just taken our 8 month old on a 6-flight family tour over the Christmas holidays without booking a seat for her. For the domestic flights <1hr, we just booked as normal and she sat on our lap. For the 9.5hr Vancouver>London flights, we booked bulkhead seats to get extra floor space. The outgoing flight was very quiet and the staff moved my wife & baby to a row with empty seats anyway, lots of sleep for everyone. For the full return flight, there was enough floor space for us to make her a bed between our legs with blankets & pillows, not great but lots better than a standard row.
Time the feeds so that they are due a bottle by take-off, drinking should help them equalise their ears. Same for descent, or at least have a dummy available.
Also check about prams if you plan to take one up to the gate. Some airlines will let you take any old thing up to the gate, others are a bit pedantic and insist its a small umbrella type.