Awards panels will soon be drawing up their long lists, but The Father by Florian Zeller (translated from the French by Christopher Hampton) and Kenneth Cranham's beautiful performance as an 80 year-old Parisian, André, who is losing his mind and his memory, will still be in the frame at the business end of the season.

James Macdonald's production, impeccably designed by Miriam Buether (set), Guy Hoare (lighting) and Christopher Shutt (sound), has moved stealthily from the Theatre Royal, Bath, via the Tricycle, Kilburn, into the West End, with no loss of snap, crackle and old pop in his pyjamas.

The snap is the flint hard dialogue as André shuffles about in his daughter Anne's flat (he lives there but thinks it's his place) upsetting the carers and her boyfriend. The crackle is in the black-outs, fused lights and fractured shards of Bach on keyboard between each scene.

And old pop – drifting among the confused (in his mind) identities of others between Paris, London and a minimal white room overlooking the park – plays a blinder as this suburban Lear who, as in Shakespeare, knows not where he did lodge last night. Cranham doesn't "act" old or doddery – he's a fairly robust 70 year-old in a trim white beard – and gives a remarkable, unaffected display of utter stillness, gravity and simmering rage and frustration.

Zeller wrote the role three years ago for one of France's legendary classical actors, Robert Hirsch (a brilliant Richard III in a long-ago World Theatre Season at the Aldwych), who's now 90. What's so moving about Cranham is that his strength of body and spirit is first afflicted then sapped by this ungovernable memory loss. He can't find his watch. One daughter has died. The other daughter, Anne (Claire Skinner, faultless in pitch, discretion and intonation), has left her husband for Pierre (Nicholas Gleaves) in London. One carer has left after André hit her with a curtain rod.

Cranham plays around the edge of all this with a clownish, unaffected twinkle, his voice set somewhere between a rasp and a wheedle, suddenly tap-dancing for the new carer (Kirsty Oswald), consuming his pills like a music hall conjuror, tracing a gesture, or a joke, like the shadow of a comedian, an echo of Max Wall.

Zeller's play mixes up the chronology and muddles André's confusion with the conflicting memorial accounts the others supply in the style of Pinter's Old Times. In this respect, it's all an unfathomable riddle, which is frustrating until you submit to the lack of plot logic. And the skill of the play comes in that apparent fault being its emotional strength.

There's one scene of moderate, but shocking, violence. And seismic misapprehensions surround the identities of a man and a woman (a carefully blank Jim Sturgeon and an outstandingly effective Rebecca Charles) who are, respectively, and possibly, Anne's husband and a doctor, and André's daughter/carer/ nurse, his lost Cordelia.

For in its study of André's condition, so common to us all at first or second hand, and increasingly so with age, Zeller, Hampton and Cranham have achieved in just eighty-five minutes of playing time something to rival and supplement the great work of Peter Brook and Pinter with the late explanatory neurologist Oliver Sacks. The Father's a play you have to see to know yourself, and those you love, better.

The Father runs at Wyndham's Theatre until 21 November 2015